ing theoretic properties of partial skew groupoid rings with applications to Leavitt path algebra

Let $\alpha=({{A}}_g,{\alpha}_g)_{g\in \operatorname{G}}$ be a group-type partial action of a connected groupoid $\operatorname{G}$ on a ring $A=\bigoplus_{z\in \operatorname{G}_0}A_z$ and $B:=A\star_{{\alpha}}\operatorname{G}$ the corresponding partial skew groupoid ring. In the first part of this paper we investigate the relation of several ring theoretic properties between $A$ and $B$. For the second part, using that every Leavitt path algebra is isomorphic to a partial skew groupoid ring obtained from a partial groupoid action $\lambda$, we characterize when $\lambda$ is group-type. In such a case, we obtain ring theoretic properties of Leavitt path algebras from the results on general partial skew groupoid rings. Several examples that illustrate the results on Leavitt path algebras are presented.

\#1<span><span>**(\*\*\* \#1 \*\*\*)**</span> </span>

\#1

[^1] [^2]

Introduction {#intro}
============

Let $\operatorname{G}$ be a groupoid and $A$ an associative and unital ring. For a partial action ${\alpha}$ of $\operatorname{G}$ on $A$ corresponds the partial skew groupoid ring $B:=A\star_\alpha \operatorname{G}$ (cf. Definition \[def-partial-skew\]). The partial skew groupoid ring is a generalization of both partial skew group ring and groupoid algebra. Precisely, if $\operatorname{G}$ is a group then $B$ is the partial skew group ring while if $\alpha$ is the trivial partial action $\alpha_g={{\rm id}}_A$, for all $g\in G$, then $B$ is the usual groupoid algebra.

Partial skew groupoid rings were firstly considered in @BFP for ordered groupoids and, after that, in @BP for general groupoids. Basic properties of $B$ were given in @BFP [@BP]. For instance, if ${\alpha}=({{A}}_g,{\alpha}_g)_{g\in \operatorname{G}}$ then $B$ is unital if and only if $A_z$ is unital, for all $z\in \operatorname{G}_0$ and $\operatorname{G}_0$ is finite, where $\operatorname{G}_0$ denotes the set of objects of $\operatorname{G}.$ Also, if $A_g$ is a unital ring for each $g\in \operatorname{G}$, then $B$ is an associative ring. Partial skew groupoid rings appear naturally in the partial Galois theory of groupoids @BP and in the context of groupoid graded rings @NyOP2020. Also, it was proved in @GY that every Leavitt path algebra is a partial skew groupoid ring.

A groupoid $\operatorname{G}$ is a disjoint union of its connected components which are full connected subgroupoids of $\operatorname{G}$. We use this decomposition of $\operatorname{G}$ to reduce partial groupoid actions to the connected case; see Remark 3.3 of @BPP. The structure of a connected groupoid is well-known. Any connected groupoid $\operatorname{G}$ is isomorphic to $\operatorname{G}_{0}^2\times \operatorname{G}(x)$, being $\operatorname{G}(x)$ the isotropy group of an object $x\in \operatorname{G}_0$. If $A=\bigoplus_{y\in \operatorname{G}_0}A_y$ and $\alpha$ is a group-type partial action (see Definition \[def-group-type\]) then the factorization of $\operatorname{G}$ induces a factorization of $B$. In fact, it was proved in @BPP that there are a global action $\beta$ of $\operatorname{G}_0^2$ on $A$ and a partial action $\gamma$ of $\operatorname{G}(x)$ on $A\ast_{\beta}\operatorname{G}_0^2$ such that $B\simeq \big(A\ast_{\beta}\operatorname{G}_0^2\big)\ast_{\gamma}\operatorname{G}(x)$. In other words, if $\alpha$ is group-type then the partial skew groupoid ring $B$ is a partial skew group ring.

In the first part of this paper we investigate ring theoretic properties of $B$ for the case that it admits the factorization $B\simeq \big(A\ast_{\beta}\operatorname{G}_0^2\big)\ast_{\gamma}\operatorname{G}(x)$. The properties of $B$ that are studied are the following: noetherianity, von Neumann regularity, the Jacobson radical, perfect, semiprimary and Krull dimension. One of the strategies used to prove the results is to apply the ring theoretic properties proved in @BC to $B$ since it is a skew partial group ring.

In the second part, we work with Leavitt path algebras. Precisely, let $L_{\Bbbk}(E)$ be the Leavitt path algebra associated to a directed graph $E$ over a field $\Bbbk.$ By @GY, we have that $L_{\Bbbk}(E)\simeq D(X)\star_{\lambda}\operatorname{G}(E)$, where $D(X)$ is $\Bbbk$ algebra and $\lambda$ is a partial action of the free path groupoid $\operatorname{G}(E)$ on $D(X)$. A general characterization for $\lambda$ to be group-type is given in Theorem \[gtype\]. A refinement of this result is given in Proposition \[daniel\] for the case that $E$ has a sink, $E^0$ is finite and $|E^{0}|\geq 2$. Assuming that $\lambda$ is group-type, we present ring theoretic properties of $L_{\Bbbk}(E)$. Examples to illustrate the results on Leavitt path algebras are also presented.

The paper is organized as follows. The background on groupoids and partial groupoid actions will be given in Section 2. The results on ring theoretic properties of the partial skew groupoid ring are presented in Section 3. Particularly, for noetherianity and regularity properties it is convenient to establish first some general results of groupoid graded rings. For this, we introduce the notion of von Neumann groupoid graded ring and, in Proposition \[gvng\], we relate this concept with von Neumann regularity. The results related to Leavitt path algebras are given in Section 4.

Conventions {#subsec:conv .unnumbered}
-----------

Throughout this work, by ring we mean an associative ring with identity. A ring with identity will be called unital. The Jacobson radical of a ring $A$ will be denoted by $J(A)$. The group of invertible elements of a unital ring $A$ is denoted by $\mathcal{U}(A)$ and the characteristic of a field $\Bbbk$ by $\operatorname{ch}(\Bbbk)$.

Preliminaries
=============

 On connected groupoids
-----------------------

A <span>*groupoid*</span> $\operatorname{G}$ is a small category in which every morphism is an isomorphism. The set of objects of $\operatorname{G}$ is denoted by $\operatorname{G}_0$. The <span>*source*</span> and <span>*the target maps*</span> $s,t:\operatorname{G}\to \operatorname{G}_0$ are defined by $s(g)=x$ and $t(g)=y$, for any morphism $g:x\to y$ of $\operatorname{G}$. The set $\operatorname{G}(x,y)$ contains the morphisms $g$ of $\operatorname{G}$ for which $s(g)=x$ and $t(g)=y$. Every object $x$ of $\operatorname{G}$ is identified with the identity morphism, that is, $x={{\rm id}}_x$. Thus $\operatorname{G}_0\subseteq \operatorname{G},$ $s(g)=g^{-1}g$ and $t(g)=gg{{}^{-1}}$, for all $g\in \operatorname{G}$. For each $x\in \operatorname{G}_0$, the <span>*isotropy group*</span> associated to $x$ is $\operatorname{G}(x):=\operatorname{G}(x,x)$ and we will fix $\operatorname{G}(x,\,):=\{g\in \operatorname{G}\,:\,s(g)=x\}$ and $\operatorname{G}(\,,x):=\{g\in \operatorname{G}\,:\,t(g)=x\}$.

We denote the composition of morphisms of a groupoid by concatenation. Notice that, for $g,h\in \operatorname{G}$, there exists $gh$ if and only if $t(h)=s(g)$. In this case, we say that $g$ and $h$ are composable. We denote by $\operatorname{G}_s\times _t\!\operatorname{G}$ the subset of $\operatorname{G}\times \operatorname{G}$ of composable pairs, that is, $\operatorname{G}_s\times _t\!\operatorname{G}$ is the pullback of $s$ and $t$ in the category of groupoids. Observe that $s(gh)=s(h)$ and $t(gh)=t(g)$, for all $(g,h)\in \operatorname{G}_s\times _t\!\operatorname{G}$.

A groupoid $\operatorname{G}$ is called <span>*connected*</span> if $\operatorname{G}(x,y)\neq \emptyset$, for any $x,y\in \operatorname{G}_0$. For an arbitrary groupoid $\operatorname{G}$ the equivalence relation on $\operatorname{G}_0$, given by $x\sim y$ if and only if $\operatorname{G}(x,y)\neq \emptyset$, induces a decomposition of $\operatorname{G}$ as a disjoint union of connected subgroupoids. Explicitly, for each equivalence class $X\subset\operatorname{G}_0$ corresponds the full subgroupoid $\operatorname{G}_X$ of $G$ whose set of objects is $X$. Hence $\operatorname{G}$ is the disjoint union of the subgroupoids $\operatorname{G}_X$, with $X\in \operatorname{G}_0/\!\!\sim$.

In order to present the structure of a connected groupoid, we recall that the <span>*coarse groupoid*</span> $Y^2$ (or Brandt groupoid) associated to any non-empty set $Y$ is the groupoid whose objects are the set $Y$ and satisfies $s(y,y')=y$, $t(y,y')=y'$ and $(y',y'')(y,y')=(y,y'')$. The next result is well known and its proof will be omitted; see, for instance, Proposition 2.1 of @BPP.

\[group:connec\] Let $\operatorname{G}$ be a connected groupoid. Then $\operatorname{G}\simeq \operatorname{G}_0^2\times \operatorname{G}(x)$ as groupoids.

Partial groupoid action
-----------------------

Throughout this subsection, $\operatorname{G}$ denotes a groupoid and $A$ denotes a ring. We start by recalling the definition of partial action given in @BP.

\[def-groupoid-action\]

A *partial action* of $\operatorname{G}$ on ${{A}}$ is a set of pairs ${\alpha}=({{A}}_g,{\alpha}_g)_{g\in \operatorname{G}}$ that satisfies:

1.  ${{A}}_g$ is an ideal of ${{A}}_{t(g)}$, ${{A}}_{t(g)}$ is an ideal of ${{A}}$ and ${\alpha}_g:{{A}}_{g{{}^{-1}}}\to {{A}}_g$ is a ring isomorphism, for all $g\in \operatorname{G}$,

2.  $\alpha_x={{\rm id}}_{A_x}$, for all $x\in \operatorname{G}_0$,

3.  ${\alpha}_h^{-1}({{A}}_{g^{-1}}\cap{{A}}_h)\subset {{A}}_{(gh)^{-1}}$, for all $(g,h)\in \operatorname{G}_s\times _t\!\operatorname{G}$,

4.  ${\alpha}_{gh}(a)={\alpha}_g{\alpha}_h(a), \, \text{ for all }\, a\in {\alpha}_h^{-1}({{A}}_{g^{-1}}\cap{{A}}_h)$, $(g,h)\in \operatorname{G}_s\times _t\!\operatorname{G}$.

We also recall other definitions that will be used later.

\[def-specific\]

A partial action ${\alpha}=({{A}}_g,{\alpha}_g)_{g\in \operatorname{G}}$ of a groupoid $\operatorname{G}$ on a ring $A$ is called

1.  <span>*global*</span>, if ${\alpha}_g{\alpha}_h={\alpha}_{gh}$, for all $(g,h)\in\operatorname{G}_s\times _t\!\operatorname{G}$,

2.  <span>*unital*</span> if each $A_g$ is a unital ring, that is, there exists a central idempotent element $1_g$ in $A$ such that $A_g=A1_g$, for all $g\in \operatorname{G}$,

3.  <span>*finite-type*</span>, if for any $z\in \operatorname{G}_0$ there are $g_1,\ldots, g_n\in \operatorname{G}(\,,z)$ such that $A_{t(g)}=\sum_{i=1}^n A_{gg_i},$ for any $g\in \operatorname{G}(z,\,).$

For the convenience of the reader, we recall Lemma 1.1 of @BP which gives some useful properties of partial groupoid actions.

\[lem:BP\] Let ${\alpha}=({{A}}_g,{\alpha}_g)_{g\in \operatorname{G}}$ be a partial action of a groupoid $\operatorname{G}$ on a ring $A$. The following statements are true:

1.  ${\alpha}_{g{{}^{-1}}}={\alpha}{{}^{-1}}_g$, for all $g\in \operatorname{G}$,

2.  ${\alpha}_g({{A}}_{g{{}^{-1}}}\cap {{A}}_h)={{A}}_{g}\cap {{A}}_{gh}$, for all $(g,h)\in \operatorname{G}_s\times _t\!\operatorname{G}$,

3.  $\alpha$ is global if and only if $A_g=A_{t(g)}$, for all $g\in \operatorname{G}$.

\[obs-pag\]<span>Let ${\alpha}=({{A}}_g,{\alpha}_g)_{g\in \operatorname{G}}$ be a partial action of a groupoid $\operatorname{G}$ on a ring $A$. Note that ${\alpha}$ induces a partial action ${\alpha}_{\operatorname{G}(x)}=({{A}}_h,{\alpha}_h)_{h\in\operatorname{G}(x)}$ of the isotropy group $\operatorname{G}(x)$ on the ring ${{A}}_x$, for each $x\in\operatorname{G}_0$. Moreover, using the decomposition of $\operatorname{G}=\dot\cup_{X\in \operatorname{G}_0/\!\sim}\operatorname{G}_X$ in connected components, it follows by Proposition 2.3 of @BPi that partial actions of $\operatorname{G}$ on $A$ are uniquely determined by partial actions of the connected groupoids $\operatorname{G}_X$, $X\in \operatorname{G}_0/\!\!\sim$, on $A$. </span>

Let $\operatorname{G}$ be a connected groupoid and $x\in \operatorname{G}_0$. Consider the following equivalence relation on $\operatorname{G}(x,\,)$: $$g\sim_x l\,\,\ \text{ if and only if }\,\, t(g)=t(l).$$ A transversal $\tau(x)=\{\tau_{y}\mid y\in \operatorname{G}_0\}$ for $\sim_x$ such that $\tau_x=x$ will be called a *transversal for $x$*. Observe that $\tau_y\in\operatorname{G}(x,y)$, for all $x\neq y\in\operatorname{G}_0$, and ${\tau}_x=x\in \operatorname{G}(x)$. The next definition was given in subsection 3.2 of @BPP.

\[def-group-type\] <span>A partial action ${\alpha}=(A_g,{\alpha}_g)_{g\in \operatorname{G}}$ of a connected groupoid $\operatorname{G}$ on a ring $A$ will be called <span>*group-type*</span> if there exist $x\in \operatorname{G}_0$ and a transversal $\tau(x)=\{\tau_{y}\mid y\in \operatorname{G}_0\}$ for $x$ such that $$\begin{aligned}
\label{cond1} 
            A_{\tau{{}^{-1}}_y}=A_x \ \ \text{and} \ \ A_{\tau_y}=A_y, \ \ \text{ for all } \ y\in\operatorname{G}_0.
    \end{aligned}$$</span>

\[obs-not-depend\]

1.  By Lemma \[lem:BP\] (iii), any global groupoid action is group-type. Also, the notion of group-type partial action does not depend on the choice of object $x$, cf. Remark 3.4 of @BPP. In fact, if ${\tau}(x)$ is a transversal for $x$ and $y\in \operatorname{G}_{0}$ then $\gamma(y):=\{\tau_z\tau{{}^{-1}}_y\mid z\in \operatorname{G}_0\}$ is a transversal for $y$.

2.  The name group-type is suggested by Corollary 3.6 of @BaPaPi in which it is observed that this kind of partial actions can be obtained from partial group actions.

Assume that ${\alpha}=(A_g,{\alpha}_g)_{g\in \operatorname{G}}$ is a group-type partial action of a connected groupoid $\operatorname{G}$ on a ring $A$. From follows $A_{g{{}^{-1}}}=A_{g{{}^{-1}}}\cap A_{s(g)}= A_{g{{}^{-1}}}\cap A_{{\tau}_{s(g)}}$ which implies that $A_g={\alpha}_g(A_{g{{}^{-1}}})= {\alpha}_g(A_{g{{}^{-1}}}\cap A_{{\tau}_{s(g)}})=A_g\cap A_{g{\tau}_{s(g)}}$, for any $g\in \operatorname{G}$. Hence $$\begin{aligned}
\label{eq1}
    A_g\subseteq A_{g\tau_{s(g)}},\,\,\, g\in \operatorname{G}.\end{aligned}$$

\[ob-invar-ideal\] <span>Let ${\alpha}$ be a partial action of a groupoid $\operatorname{G}$ on a ring $A$. Consider an $\alpha$-invariant ideal $I$ of $A$, that is, ${\alpha}_g(I\cap A_{g{{}^{-1}}})\subseteq I$, for all $g\in \operatorname{G}$. In this case the family ${\alpha}|_I=(I_g, {\alpha}_g)_{g\in \operatorname{G}}$, where $I_g:=I\cap A_{g}$, is a partial action of $\operatorname{G}$ on $I$. It is clear that if ${\alpha}$ is group-type then ${\alpha}|_I$ is group-type.</span>

 The partial skew groupoid ring
-------------------------------

Throughout this subsection, ${\alpha}=(A_g,{\alpha}_g)_{g\in \operatorname{G}}$ denotes a unital partial action of a groupoid $\operatorname{G}$ on a ring $A$. We will assume that $A_g=A1_g$, where $1_g$ is a central idempotent of $A$, for all $g\in\operatorname{G}$.

\[def-partial-skew\] <span>The <span>*partial skew groupoid ring*</span> $A\star_{\alpha}\operatorname{G}$ associated to $\alpha$ is the set of formal sums $\sum_{g\in \operatorname{G}}a_g\delta_g$, where $a_g\in A_g$, with the usual addition and multiplication induced linearly by the following rule $$\label{prod}
            (a_g\delta_g)(a_h\delta_h)=
            \begin{cases}
                a_g\alpha_g(a_h1_{g^{-1}})\delta_{gh}, &\text{if $(g,h)\in \operatorname{G}_s\times _t\!\operatorname{G}$,}\\
                0, &\text{otherwise},
        \end{cases}$$ for all $g, h\in \operatorname{G}$, $a_g\in A_g$ and $a_h\in A_h$.</span>

The partial skew groupoid ring $A\star_{\alpha}\operatorname{G}$ is an associative ring. It was proved in Section 3 of @BFP that if $\operatorname{G}_0$ is finite then $A\star_{\alpha}\operatorname{G}$ is unital with identity $1_{A\star_{\alpha}\operatorname{G}}=\sum_{y\in\operatorname{G}_0}1_y\delta_y$. On the other hand it was shown in @BPP that, if $\operatorname{G}$ is connected then the factorization of $\operatorname{G}$ given by Proposition \[group:connec\] induces, under suitable conditions, a factorization of the partial skew groupoid ring. In order to review this construction, assume that $\operatorname{G}$ is connected, $\operatorname{G}_0$ is finite, ${\alpha}$ is group-type and $A=\bigoplus_{z\in \operatorname{G}_0}A_{z}$. We also fix $x\in \operatorname{G}_0$ and $\tau(x)=\{\tau_{y}\mid y\in \operatorname{G}_0\}$ a transversal for $x$ such that is satisfied.

For each $u=(y,z)\in \operatorname{G}_0^2$, we consider $$\begin{aligned}
\label{action-beta}
    &B_u=A_{t(u)}=A_z,& &{\beta}_u={\alpha}_{{\tau}_{t(u)}}{\alpha}_{{\tau}{{}^{-1}}_{s(u)}}:A_y\to A_z.& \end{aligned}$$ By Lemma 4.1 of @BPP, the family ${\beta}=(B_u,{\beta}_u)_{u\in\operatorname{G}_0^2}$ is a global action of $\operatorname{G}_0^2$ on $A$. Hence we may consider the skew groupoid ring $C:=A\star_ {{\beta}}\operatorname{G}_0^2$. There is a partial action $\gamma$ of $\operatorname{G}(x)$ on $C$ defined in the following way. Given $z\in \operatorname{G}_0$ and $h\in \operatorname{G}(x)$, set $C_{z,h}:={\alpha}_{{\tau}_z}(A_h)$ and $$\begin{aligned}
\label{Ch}
    C_h:=\bigoplus_{u\in\operatorname{G}_0^2}C_{t(u),h}\delta_u=\bigoplus_{u\in\operatorname{G}_0^2}{\alpha}_{{\tau}_{t(u)}}(A_h)\delta_u.\end{aligned}$$ Also, $\gamma_{z,h}:C_{z,h{{}^{-1}}}\to C_{z,h}$ is defined by ${\alpha}_{{\tau}_z}(a)\mapsto {\alpha}_{{\tau}_z}({\alpha}_h(a))$, for all $a\in A_{h{{}^{-1}}}$. These maps induce the following ring isomorphism $$\begin{aligned}
\label{gamma-h}
    \gamma_h:C_{h{{}^{-1}}}\to C_h,\quad  \gamma_h({\alpha}_{{\tau}_{t(u)}}(a)\delta_u)= \gamma_{t(u),h}(a)\delta_u, \,\,\, a\in A_{h{{}^{-1}}},\,\, u\in \operatorname{G}_0^2.\end{aligned}$$ From Lemmas 4.2 and 4.3 of @BPP follow that $\gamma=(C_h, \gamma_h)_{h\in \operatorname{G}(x)}$ is a unital partial action of $\operatorname{G}(x)$ on $C$. Moreover $C_h=C1'_h$, where $1'_h=\sum_{z\in \operatorname{G}_0}{\alpha}_{\tau_z}(1_h)\delta_{(z,z)}$, for all $h\in \operatorname{G}(x)$. In order to enunciate Theorem 4.4 of @BPP, which give us the factorization of $A\star_{\alpha}\operatorname{G}$, we consider the notation: $g_x={\tau}{{}^{-1}}_{t(g)}g{\tau}_{s(g)}\in \operatorname{G}(x)$, for all $g\in \operatorname{G}$.

\[teo-decomp\] Suppose that $\operatorname{G}$ is connected, $\operatorname{G}_0$ is finite, ${\alpha}$ is group-type and $A=\bigoplus_{z\in \operatorname{G}_0}A_{z}$. Then the map $\psi:A\star_{\alpha}\operatorname{G}\to (A\star_{{\beta}}\operatorname{G}_0^2)\star_ \gamma\operatorname{G}(x)$ given by $$\label{varphi} \psi(a\delta_g)=a\delta_{(s(g),t(g))}\delta_{g_x},\quad  a_g\in A_g,\,\, g\in \operatorname{G},$$ is a ring isomorphism.

The factorization of $A\star_{\alpha}\operatorname{G}$ given above will be useful in the rest of the paper. We end the background with the following.

\[latoga\] Suppose that ${\alpha}$ is group-type. If $\alpha$ is finite-type, then $\gamma$ defined in is finite-type.

Since $\alpha$ is finite-type, there are $g_1, \dots, g_n\in \operatorname{G}(\,,x)$ such that $A_x=\sum_{i=1}^n A_{gg_i}$. For each $h\in \operatorname{G}(x)$ we take $\{hg_1\tau_{s(gg_1)}, \ldots, hg_n\tau_{s(gg_n)}\}\subseteq \operatorname{G}(x)$. Then $$\begin{aligned}
        A\star_{{\beta}}\operatorname{G}_0^2&=\bigoplus_{u\in \operatorname{G}_0^2}A_{t(u)}\delta_u= \bigoplus_{u\in \operatorname{G}_0^2}\alpha_{\tau_{t(u)}}(A_x)\delta_u=\bigoplus_{u\in \operatorname{G}_0^2}\sum_{i=1}^n\alpha_{\tau_{t(u)}}(A_{hg_i})\delta_u\\
        &=\sum_{i=1}^n\bigoplus_{u\in \operatorname{G}_0^2}\alpha_{\tau_{t(u)}}(A_{hg_i})\delta_u\stackrel{\eqref{eq1}}\subseteq\sum_{i=1}^n\bigoplus_{u\in \operatorname{G}_0^2}\alpha_{\tau_{t(u)}}(A_{hg_i\tau_{s(hg_i)}})\delta_u\\
        &\stackrel{\eqref{Ch}}=\sum_{i=1}^nC_{hg_i\tau_{s(hg_i)}}\subseteq A\star_{{\beta}}\operatorname{G}_0^2.
    \end{aligned}$$ Hence $A\star_{{\beta}}\operatorname{G}_0^2=\sum\limits_{i=1}^nC_{gg_i\tau_{s(gg_i)}}$ and we conclude that $\gamma$ is finite-type.

Ring theoretic properties of $A\star_{\alpha}\operatorname{G}$ {#sec-3}
==============================================================

In all what follows in this section, $\operatorname{G}$ is a connected groupoid such that $\operatorname{G}_0$ is finite and ${\alpha}=(A_g,{\alpha}_g)_{g\in \operatorname{G}}$ is a group-type unital partial action of $\operatorname{G}$ on $A=\bigoplus_{z\in \operatorname{G}_0} A_z$. We will assume that there is $x\in \operatorname{G}_0$ and $\tau(x)=\{\tau_{y}\,:\,y\in\operatorname{G}_0\}$ a transversal for $x$ that satisfies and $A_g=A1_g$, where $1_g$ is a central idempotent of $A$, for all $g\in\operatorname{G}$. We also assume that $\beta$ and $\gamma$ are as in the previous section.

The aim in this section is to establish relations between ring theoretic properties of $A$ and $A\star_{\alpha}\operatorname{G}$. First, observe that $$\begin{aligned}
\label{inclusion-map}
    \varphi: A\to A\star_\alpha \operatorname{G}\,\text{ given by }\,  \varphi(a)=\sum_{z\in \operatorname{G}_0}(a1_z)\delta_z,\quad a\in A,    \end{aligned}$$ is an injective ring homomorphism and whence we identify $A$ as a unital subring of $A\star_\alpha \operatorname{G}$. Moreover $\varphi$ induces the following ring isomorphisms $A\simeq  \oplus_{z\in \operatorname{G}_0}A_z\delta_z\simeq \oplus_{w\in (\operatorname{G}^2_0)_0}A_w\delta_{w}$. Consequently $$\label{isosos}A\simeq (A\star_{\alpha}\operatorname{G})_0 \simeq  (A\star_{{\beta}}\operatorname{G}_0^2)_0.$$

Noetherianity
-------------

The relation of the artinian property between $A$ and $A\star_{\alpha}\operatorname{G}$ was given in Theorem 1.3 of @NOP, for an arbitrary groupoid $\operatorname{G}$. In our context, this result was refined in Theorem 5.11 of @BPP. Here we explore the relation of the noetherian property between $A$ and $A\star_{\alpha}\operatorname{G}$.

We recall some notions and facts on group and groupoid graded rings. We say that that a ring $R$ is called <span>*$\operatorname{G}$-graded*</span> if there is a set $\{ R_g \}_{g \in \operatorname{G}}$ of additive subgroups of $R$ such that $R = \bigoplus_{g \in \operatorname{G}} R_g$, $R_gR_h\subset R_{gh}$ if $(g,h)\in \operatorname{G}_s\times _t\!\operatorname{G}$ and $R_g R_h = \{ 0 \}$ otherwise. A $\operatorname{G}$-graded ring $R=\bigoplus_{g \in \operatorname{G}} R_g$ that satisfies $R_g R_h = R_{gh}$ for all $(g,h)\in \operatorname{G}_s\times _t\!\operatorname{G}$ is said <span>*strongly graded*</span>. Then by setting $(A\star_{\alpha}\operatorname{G})_g=A_g\delta_g$ for any $ g\in \operatorname{G},$ follows from that $A\star_{\alpha}\operatorname{G}$ is a $\operatorname{G}$-graded ring. Moreover, by Proposition 3.4 of @BFP, $A\star_{\alpha}\operatorname{G}$ is strongly graded if and only if $\alpha$ is global. If $\operatorname{G}=G$ is a group we say that $G$-graded ring $R = \oplus_{g \in G} R_g$ is <span>*epsilon-strongly graded*</span> if $R_g R_h = R_g R_{g^{-1}} R_{gh} = R_{gh} R_{h^{-1}} R_h,\quad \text{for all }g,h\in G.$ It is not difficult to see that unital partial crossed products are epsilon-strongly graded, see Theorem 35 of @NyOP2018 for details.

\[noet\] Let $\operatorname{G}$ be a finite groupoid, $R=\bigoplus_{g\in \operatorname{G}}R_g$ be a strongly $\operatorname{G}$-graded unital ring and $R_0=\bigoplus_{z\in \operatorname{G}_0}R_z$. If $R_0$ is left (right) noetherian, then $R$ is left (right) noetherian.

Since $R_g R_{g{{}^{-1}}}=R_{t(g)}$, for all $g\in \operatorname{G}$, we obtain a $(R_g,R_{g{{}^{-1}}})$-bimodule epimorphism $\mu: R_g \otimes_{R_{s(g)}}R_{g{{}^{-1}}}\to R_{t(g)}.$ Similarly, there exists a $(R_{g{{}^{-1}}},R_{g})$-bimodule epimorphism $\tau: R_{g{{}^{-1}}} \otimes_{R_{t(g)}}R_{g}\to R_{s(g)}$. Then, the Morita context $(R_{t(g)}, R_{s(g)}, R_g, R_{g{{}^{-1}}}, \mu, \tau)$ is strict. Thus ${R_g}$ is a left projective and finitely generated $R_{t(g)}$-module and ${R_g}$ is a right projective and finitely generated $R_{s(g)}$-module. Consequently, ${R_g}$ is a left projective and finitely generated right $R_0$-module . Using that $\operatorname{G}$ is finite we conclude that $R$ is a left and right finitely generated $R_0$-module. The fact that $R_0$ is left (right) noetherian implies that ${R}$ is a left and right noetherian $R_0$-module. Finally, from $R_0\subseteq R$ we conclude that $R$ is a left and right noetherian $R$-module and the result follows.

In order to prove the main result of this subsection, we recall the following. A group $G$ is called a <span>*polycyclic-by-finite*</span> if there exists a subnormal series $$\{1\}=G_0\unlhd G_1\unlhd \ldots\unlhd G_n\unlhd G_{n+1}=G$$ such that $G/G_n$ is finite and $G_{i+1}/G_i$ is cyclic, for all $0\leq i\leq n-1$.

\[noete\] The following assertions hold.

1.  If $A\star_{\alpha}\operatorname{G}$ is left noetherian then $A$ and $A\star_{{\beta}}\operatorname{G}_0^2$ are left noetherian.

2.  If $A$ is left noetherian and $\operatorname{G}(x)$ is a polycyclic-by-finite group then $A\star_{\alpha}\operatorname{G}$ is left noetherian.

\(i) From Theorem \[teo-decomp\] and Proposition 3.4 of @NOP follow that $A\star_{{\beta}}\operatorname{G}_0^2$ is left noetherian. Applying again Proposition 3.4 of @NOP for $A\star_{{\beta}}\operatorname{G}_0^2$ we conclude that $A$ is left noetherian.

\(ii) By Lemma \[noet\] the ring $C:=A\star_{{\beta}}\operatorname{G}_0^2$ is left noetherian. Since $C\star_ \gamma\operatorname{G}(x)$ is epsilon strongly graded Theorem 3.7 of @L implies that it is left noetherian and whence the result follows from Theorem \[teo-decomp\].

Von Neumann regularity
----------------------

Recall that a unital associative ring $R$ is von Neumann regular if $a\in aRa,$ for all $a\in R$. There are several equivalent conditions to the notion of von Neumann regularity. Here, we will use the following equivalences: $R$ is von Neumann regular $\Leftrightarrow$ if every principal left ideal is a direct summand of the left $R$-module $R$ $\Leftrightarrow$ if every finitely generated left ideal is a direct summand of the left $R$-module $R$.

The next definition is inspired in the group case which was considered in Section 2.2 of @La2. A $\operatorname{G}$-graded ring $R=\bigoplus_{g\in \operatorname{G}} R_g $ will be called a <span>*graded von Neumann regular* </span> if and only if $a\in aRa,$ for each $a\in R_g$ and $g\in \operatorname{G}.$ Clearly, a $\operatorname{G}$-graded ring that is von Neumann regular is graded von Neumann regular. The converse is not true even in the group case; see Example 2.4 of @La2.

\[obs-g-grade\]<span> Let $B:=A\star_{\alpha}\operatorname{G}$ and $C:=A\star_{\beta}\operatorname{G}_{0}^2$. By Theorem \[teo-decomp\], we have $B\simeq C\star_{\gamma}\operatorname{G}(x)$. Notice that if $B$ is $\operatorname{G}(x)$-graded von Neumann regular then $B$ is $\operatorname{G}$-graded von Neumann regular. In fact, let $g\in \operatorname{G}$ and $a_g\delta_g\in A\star_{\alpha}\operatorname{G}$. By , $\psi(a_g\delta_g)\in (C\star_{\gamma}\operatorname{G}(x))_{g_x}$. Then $\psi(a_g\delta_g)\in \psi(a_g\delta_g)(C\star_{\gamma}\operatorname{G}(x))\psi(a_g\delta_g)$ which implies that $a_g{\delta}_g\in (a_g\delta_g)(A\star_{\alpha}\operatorname{G}) (a_g\delta_g)$. Therefore $A\star_{\alpha}\operatorname{G}$ is $\operatorname{G}$-graded von Neumann regular.</span>

The next auxiliary result has immediate proof which will be omitted.

\[vnr\] Let $\{I_j\}_{1\leq j\leq n}$ be a family of ideals of a ring $R$ that satisfies $I_iI_j=I_jI_i=0$ for all $i\neq j$. Then $\sum_{j=1}^n I_j$ is von Neumman regular if and only if $I_j$ is von Neumman regular, for all $1\leq j\leq n$.

Now we present an extension of Theorem 3 of @HY for the context of groupoid graded rings.

\[gvng\] Let $R=\bigoplus_{g\in \operatorname{G}}R_g$ be a $\operatorname{G}$-graded ring and $R_0=\bigoplus_{z\in \operatorname{G}_0}R_z$. Then the following assertions hold.

1.  If $R$ is graded von Neumann regular then $R_0$ is von Neumann regular.

2.  If $R_0$ is von Neumann regular, $R$ is strongly $\operatorname{G}$-graded and unital, then $R$ is graded von Neumann regular.

\(i) By Lemma \[vnr\], it is enough to show that $R_z$ is von Neumann regular, for all $z\in \operatorname{G}_0$. Let $a\in R_z$. Then there is $r\in R$ such that $a=ara$. Assume that $r=\sum_{g\in \operatorname{G}}r_g$, with $r_g\in R_g$ for all $g\in \operatorname{G}$. Notice that $R_zR_z\subset R_z$. Hence $ara-ar_za=a-ar_za\in R_z$ and consequently $$ara-ar_za=\sum_{g\in \operatorname{G}\setminus\{z\}}ar_ga \in R_z\cap \left( \sum\limits_{g\in \operatorname{G}\setminus\{z\}}R_g\right)=0.$$ Thus $a=ara=ar_za\in aR_za$ and whence $R_z$ is von Neumann regular.

\(ii) Let $a\in R_g$ and $g\in \operatorname{G}$. Since $R$ is strongly $G$-graded, ${R_g}$ is a left $R_{t(g)}$-module projective and finitely generated (see the proof of Lemma \[noet\]). Now, $R_{g{{}^{-1}}}a\subseteq R_{s(g)}$ and $R_{s(g)}R_{g{{}^{-1}}}a=R_{g{{}^{-1}}}a$, that is, $R_{g{{}^{-1}}}a$ is a left ideal of $R_{s(g)}$. Since $R_{s(g)}$ is von Neumann regular by Lemma \[vnr\], there is an idempotent $u\in R_{s(g)}$ such that $R_{g{{}^{-1}}}a=R_{s(g)}u$. Note also that, by Proposition 2.1.1 of @lundstrom2004, $1_R=\sum_{z\in \operatorname{G}_0}1_z$ and $R_{s(g)}$ is a unital ring with identity element $1_{s(g)}$. Consequently $u=1_{s(g)}u=ra$, for some $r\in R_{g{{}^{-1}}}$. Moreover $R_{t(g)}a=R_gR_{g{{}^{-1}}}a=R_gR_{s(g)}u=R_gu$ which implies that $a=1_{t(g)}a=bu$ for some $b\in R_g$. Then $ara=au=bu^2=bu=a$, that is, $R$ is graded von Neumann regular.

<span>*For the rest of this subsection we will assume that $\operatorname{G}$ is finite*</span>. Recall that the trace map ${\rm tr}_\alpha:A\to A$ associated to the partial action $\alpha$ is defined by $${\rm tr}_\alpha(a)=\sum_{g\in \operatorname{G}}\alpha_g(a1_{g{{}^{-1}}}),\quad a\in A.$$ Since any group is a groupoid, the trace map is defined to partial group actions. Notice that ${\rm tr}_\alpha(1_A)=\sum_{g\in \operatorname{G}}1_{g}$. The trace map has a main role in the Galois theory for partial groupoid actions. More details can be seen in @BP.

In the sequel we prove two auxiliary results.

\[lem-jacobson1\] Let $(C,\gamma)$ be the partial action of $\operatorname{G}(x)$ on $C=A\ast_{\beta}\operatorname{G}_0^2$ given by and and $a\in A_x$. Then:

1.  ${\rm tr}_{\gamma}(1_C)=\sum_{z\in \operatorname{G}_0}{\alpha}_{\tau_z}({\rm tr}_{{\alpha}_{\operatorname{G}(x)}}(1_x))\delta_{(z,z)}$.

2.  An element $b=\sum_{z\in \operatorname{G}_0}{\alpha}_{{\tau}_z}(a)\delta_{(z,z)}\in C$ is invertible in $C$ if and only if $a$ is invertible in $A_x$. In this case, $b^{-1}=\sum_{z\in \operatorname{G}_0}{\alpha}_{{\tau}_z}(a^{-1})\delta_{(z,z)}$.

Since $C_h=C1'_{h}$ with $1'_h=\sum_{z\in \operatorname{G}_0}{\alpha}_{{\tau}_z}(1_h)\delta_{(z,z)}$ and $1_C=\sum_{y\in\operatorname{G}_0}1_y\delta_{(y,y)}$ we have $$\begin{aligned}
        {\rm tr}_\gamma(1_C)&=\sum_{h\in \operatorname{G}(x)}\sum_{z\in \operatorname{G}_0}{\alpha}_{\tau_z}(1_{h})\delta_{(z,z)}=\sum_{z\in \operatorname{G}_0}{\alpha}_{\tau_z}\Big(\sum_{h\in \operatorname{G}(x)}1_{h}\Big)\delta_{(z,z)}\\
        &=\sum_{z\in \operatorname{G}_0}{\alpha}_{\tau_z}\Big(\sum_{h\in \operatorname{G}(x)}{\alpha}_{\operatorname{G}(x)}(1_x1_{h{{}^{-1}}})\Big)\delta_{(z,z)}=\sum_{z\in \operatorname{G}_0}{\alpha}_{\tau_z}\big({\rm tr}_{{\alpha}_{\operatorname{G}(x)}}(1_x)\big)\delta_{(z,z)},
    \end{aligned}$$ and whence (i) follows. For (ii), assume that $a$ is invertible in $A_x$ and consider the element $c=\sum_{z\in \operatorname{G}_0}{\alpha}_{{\tau}_z}(a^{-1})\delta_{(z,z)}\in C$. Thus, $bc=\sum_{z\in \operatorname{G}_0}{\alpha}_{\tau_z}(1_x)\delta_{(z,z)}=\sum_{z\in \operatorname{G}_0}1_z\delta_{(z,z)}=1_C$ and whence $c=b^{-1}$. Conversely, consider $c\in C$ the inverse of $b$. Then, there are elements $a_u\in B_u=A_{t(u)}$ such that $c=\sum_{u\in \operatorname{G}_0^2}a_u\delta_u$. As ${\alpha}_{{\tau}_{t(u)}}:A_x\to A_{t(u)}$ is a ring isomorphism, for each $u\in \operatorname{G}_0^2$, there is $a_{x,u}\in A_x$ such that $a_u={\alpha}_{{\tau}_{t(u)}}(a_{x,u})$. Hence, $$\begin{aligned}
        \sum_{z\in \operatorname{G}_0}1_z\delta_{(z,z)}=1_C=bc&=\sum_{z\in \operatorname{G}_0}\sum_{u\in \operatorname{G}(\,\,,\,z)}{\alpha}_{{\tau}_z}(aa_{x,u})\delta_u\\&=\sum_{z\in \operatorname{G}_0}\sum_{y\in \operatorname{G}_0}{\alpha}_{{\tau}_z}(aa_{x,(y,z)})\delta_{(y,z)}.
    \end{aligned}$$ Thus, $a_{x,(y,z)}=0$ if $y\neq z$ and $aa_{x,(z,z)}=1_x$. Then $a_{x,(z,z)}=a^{-1}$ and we have that $c=\sum_{z\in \operatorname{G}_0}{\alpha}_{{\tau}_z}(a^{-1})\delta_{(z,z)}$.

\[lem-jacobson2\] Let $(A,\beta)$ be the global action of $\operatorname{G}_0^2$ on $A$ given in the previous section. Then ${\rm tr}_\beta(1_A)=\vert \operatorname{G}_0 \vert 1_A$.

It is clear that $$\begin{aligned}
        {\rm tr}_\beta(1_A)&= \sum\limits_{u\in \operatorname{G}_0^2 }\beta_u(1_{s(u)})=\sum\limits_{u\in \operatorname{G}_0^2 }1_{t(u)}= \sum\limits_{(y,z)\in \operatorname{G}_0^2 }1_{z}=\vert \operatorname{G}_0 \vert\sum\limits_{z\in \operatorname{G}_0 }1_{z}=\vert \operatorname{G}_0 \vert 1_A,
    \end{aligned}$$ because $A=\bigoplus\limits_{z \in \operatorname{G}_0}A_z$.

Now we can prove the main result of this subsection.

\[vn1\] Assume that $\operatorname{G}$ is finite and let $\beta$ be the global action of $\operatorname{G}$ on $A$ given in . Then the following assertions hold.

1.  If $A$ is von Neumann regular and $I$ is a left principal ideal of $A\star_{{\beta}}\operatorname{G}_0^2$, then $I$ is an $A$-direct summand of $A\star_{{\beta}}\operatorname{G}_0^2$.

2.  If $|\operatorname{G}_0|1_A$ is invertible in $A$ then the following statements are equivalent:

    1.  $A$ is von Neumann regular,

    2.  $A\star_{{\beta}}\operatorname{G}_0^2$ is graded von Neumann regular,

    3.  $A\star_{{\beta}}\operatorname{G}_0^2$ is von Neumann regular,

    4.  $A\star_{\alpha}\operatorname{G}$ is graded von Neumann regular.

    Moreover, under the assumption that at least one of the above statements holds and ${\rm tr}_{{\alpha}_{\operatorname{G}(x)}}(1_x)$ is invertible in $A_x$ the following additional statement holds:

    1.  $A\star_{\alpha}\operatorname{G}$ is von Neumann regular.

For (i), consider $u,v\in \operatorname{G}_0^2$ such that $s(v)=t(u)$ and $a\in B_u=A_{t(u)}$. Then $A_{t(v)}\delta_v\cdot a\delta_u=A_{t(v)}\beta_v(a1_{s(v)})\delta_{vu}$. Suppose that $I=(A\star_{{\beta}}\operatorname{G}_0^2)l$, with $l=\sum_{u\in \operatorname{G}_0^2}a_u\delta_u$. Then $$I=\sum_{u,v\in \operatorname{G}_0^2}A_{t(v)}\delta_va_u\delta_u=\sum_{t(u)=s(v)}A_{t(v)}\beta_v(a_u1_{s(v)})\delta_{vu}.$$ For each $u\in \operatorname{G}_0^2$, let $$I_u:=\sum_{s(v)=t(u)} A_{t(v)}\beta_v(a_u1_{s(v)})=\sum_{s(v)=t(u)} A1_{t(v)}\beta_v(a_u1_{s(v)}).$$ It is clear that $I_u$ is a finitely generated left ideal of $A$. Since $A$ is von Neumann, $I_{u}$ is a direct summand of the left $A$-module $A$. Hence $I$ is an $A$-direct summand of $C:=A\star_{{\beta}}\operatorname{G}_0^2$.

For (ii), using the identification map (\[inclusion-map\]) we have by that $C$ is strongly graded and $(A\star_{{\beta}}\operatorname{G}_0^2)_0=C_0=A$. Thus, the equivalence (a) $\Leftrightarrow $ (b) follows from Proposition \[gvng\]. In order to prove (a) $\Rightarrow $ (c), consider a principal left ideal $I$ of $C$. By (i), $I$ is an $A$-direct summand of $C$. Also, by Lemma \[lem-jacobson2\], $\operatorname{tr}_{\beta}(1_A)=\vert \operatorname{G}_0 \vert1_A$. Then, Theorem 5.5 of @FMP implies that $I$ is a $C$-direct summand of $C$. Thus $C$ is von Neumann regular. For (c) $\Rightarrow $ (d), observe that $C\star_{\gamma}\operatorname{G}(x)$ is an epsilon strongly $G(x)$-graded ring such that $(C\star_{\gamma}\operatorname{G}(x))_x=C$. Then, from Corollary 3.11 of @La2 we obtain that $C\star_{\gamma}\operatorname{G}(x)$ is $G(x)$-graded von Neumann regular. By Remark \[obs-g-grade\], $A\star_{\alpha}\operatorname{G}$ is $\operatorname{G}$-graded von Neumann regular. Finally, note that (d) $\Rightarrow $ (a) follows from Proposition \[gvng\] because $(A\star_{\alpha}\operatorname{G})_0=\bigoplus\limits_{z\in \operatorname{G}_0}A_z\delta_z\simeq A$. To finish the proof we suppose also that ${\rm tr}_{{\alpha}_{\operatorname{G}(x)}}(1_x)$ is invertible in $A_x$ and prove (e). Indeed, it follows from itens (i) and (ii) of Lemma \[lem-jacobson1\] that ${\rm tr}_{\gamma}(1_C)$ is an invertible element in the von Neumann regular ring $A*_{\beta}\operatorname{G}_0^2$. Hence, Theorem 4.3 of @FL implies that $(A\star_{{\beta}}\operatorname{G}_0^2)\star_ \gamma\operatorname{G}(x)$ is von Neumann regular. Thus, we obtain from Theorem \[teo-decomp\] that $A\star_{\alpha}\operatorname{G}$ is von Neumann regular.

<span>Notice that the assumption $|\operatorname{G}_0|1_A $ is invertible in $A$ in Theorem \[vn1\] was only used to prove (a) $\Rightarrow $ (c).</span>

Jacobson radical
----------------

Here we investigate the relation between $J(A)$ and $J(A*_\alpha \operatorname{G})$. The following properties of $J(A)$ will be useful later.

\[jsum\]The following statements hold.

1.  $J(A)$ is an $\alpha$-invariant ideal of $A$, that is, $\alpha_g(J(A)\cap A_{g{{}^{-1}}})\subset J(A)$ for all $g\in \operatorname{G}$.

2.  $J(A_{t(g)})1_g=J(A_g),$ for all $g\in \operatorname{G}$; in particular $J(A_g)$ is unital.

3.  $J(A)=\bigoplus_{z\in \operatorname{G}_0} J( A_z)$.

4.  ${\mathcal U}(A)={\mathcal U}(A\star_\alpha \operatorname{G}) \cap A$.

5.  $J(A)  = A\cap J(A\star_\alpha \operatorname{G})$.

Let $g\in \operatorname{G}$. As $J(A)$ is an hereditary radical and ${\alpha}_g$ is a ring isomorphism one has $$\alpha_g(J(A)\cap A_{g^{-1}})=\alpha_g(J(A_{g^{-1}}))=J(\alpha_g(A_{g^{-1}}))=J(A_g)=J(A)\cap A_g,$$ and whence (i) follows. Notice that (iii) is immediate because $A=\bigoplus_{z\in \operatorname{G}_0}A_z$. To prove (ii), observe that $A_{t(g)}=A_g\oplus A_{t(g)}(1_{t(g)}-1_g)$. Consequently, we have that $J(A_{t(g)})=J(A_g)\oplus J\big(A_{t(g)}(1_{t(g)}-1_g)\big)$ which implies that $J(A_{t(g)})1_g=J(A_g)$.

The inclusion $\subseteq$ in (iv) is clear. For the reverse, consider the $(A,A)$-bimodule epimorphism $$\label{pia}\pi_A: A\star_\alpha \operatorname{G}\to A, \quad \pi_A\Big(\sum_{g\in \operatorname{G}} a_g\delta_g\Big)=\sum_{e\in \operatorname{G}_0} a_e\delta_e.$$ Let $u\in {\mathcal U}(A\star_\alpha \operatorname{G}) \cap A$. Then there is $v\in A\star_\alpha \operatorname{G}$ such that $uv=vu=1_{A\star_\alpha \operatorname{G}}$. Since $\pi_A(u)=u$ it follows that $u\pi_A(v)=\pi_A(v)u=1_{A}$ and consequently $u\in {\mathcal U}(A)$.

\(v) Let $a=\sum_{g\in \operatorname{G}}a_g\delta_g\in A\star_\alpha \operatorname{G}$, where $a_g\in A_g$, for all $g\in \operatorname{G}$. Observe that $a_g\delta_g=(1_g\delta_g)\big({\alpha}_{g{{}^{-1}}}(a)\delta_{s(g)}\big)=(1_g\delta_g)\varphi({\alpha}_{g{{}^{-1}}}(a))\in (1_g\delta_g)A,$ being $\varphi$ the inclusion map defined in . Hence $a\in \sum_{g\in \operatorname{G}}(1_g\delta_g)A$ and $A\star_\alpha \operatorname{G}=\sum_{g\in \operatorname{G}}(1_g\delta_g)A$. Let $M$ be a simple right $A\star_\alpha \operatorname{G}$-module. Then, given a non-zero element $m\in M$, we have $M=m(A\star_\alpha \operatorname{G})=\sum_{g\in \operatorname{G}}\left(m(1_g\delta_g)\right)A$, thus $M$ is a finitely generated $A$-module. Fix $N:=MJ(A)$. We shall check that $N=\{0\}$. If $g\in \operatorname{G}$ and $t\in A$, then $$\begin{aligned}
        N(t1_g\delta_g)&=M(J(A)t)(1_g\delta_g)\subseteq MJ(A)(1_g\delta_g)\stackrel{{\rm (iii)}}{=}MJ(A_{t(g)})(1_g\delta_g)\\&\stackrel{{\rm (ii)}}{=}MJ(A_{g})(1_g\delta_g)
        \stackrel{(\ast)}{=}M(1_g\delta_g)J(A_{g{{}^{-1}}})\subseteq M(1_g\delta_g)J(A_{s(g)})\\&\subseteq MJ(A)=N.
    \end{aligned}$$ Note that $(\ast)$ is true because $b(1_g\delta_g)=(1_g\delta_g)({\alpha}_{g{{}^{-1}}}(b)\delta_{s(g)})$ and ${\alpha}_{g{{}^{-1}}}(b)\in J(A_{g{{}^{-1}}})$, for all $b\in J(A_g)$. Therefore $N(t1_g\delta_g)=N,$ for all $t\in A$ and $g \in \operatorname{G}$, which implies that $N$ is a right $A\star_\alpha \operatorname{G}$-submodule of $M$. Since $M$ is simple, $N= M$ or $N=\{0\}$. If $N=M$ then $MJ(A)=M$. Using that $M$ is an $A$-module finitely generated, we obtain from Lemma 2.4 of @P that $M=\{0\}$ which is a contradiction. Therefore $N=\{0\}$ and thus $J(A)$ annihilates every simple $A\star_\alpha \operatorname{G}$-module. Consequently $J(A)\subseteq J(A\star_\alpha \operatorname{G})$. For the reverse, we recall that $a\in J(A)$ if and only if $1-xa\in \mathcal{U}(A)$. Hence it follows from (iv) that $A\cap J(A\star_\alpha \operatorname{G})\subset J(A)$.

We end this subsection with the following.

\[jaco\]If ${\rm tr}_{{\alpha}_{\operatorname{G}(x)}}(1_x)$ is invertible in $A_x$ and $\vert G_0\vert1_A$ is invertible in $A$ then $$J(A*_\alpha \operatorname{G})\simeq J(A)*_\alpha \operatorname{G}.$$

It is immediate from Theorem \[teo-decomp\] that $J(A*_\alpha \operatorname{G})\simeq J\big((A\star_{{\beta}}\operatorname{G}_0^2)\star_ \gamma\operatorname{G}(x)\big)$. By Lemma \[lem-jacobson1\], ${\rm tr}_{\gamma}(1_C)$ is invertible in $C=A\star_{{\beta}}\operatorname{G}_0^2$. Then, it follows from Proposition 6.7 of @FL that $J((A\star_{{\beta}}\operatorname{G}_0^2)\star_ \gamma\operatorname{G}(x))=J(A\star_{{\beta}}\operatorname{G}_0^2)\star_ \gamma\operatorname{G}(x)$. Also, by Lemma \[lem-jacobson2\] and Theorem 5.7 of @FMP we have that $J(A*_\beta \operatorname{G}^2_0)=J(A)*_\beta \operatorname{G}^2_0$. Hence $$J((A\star_{{\beta}}\operatorname{G}_0^2)\star_ \gamma\operatorname{G}(x))=(J(A)\star_{{\beta}}\operatorname{G}_0^2)\star_ \gamma\operatorname{G}(x).$$ By Lemma \[jsum\] (i) and Remark \[ob-invar-ideal\] the restriction of $\alpha$ to $J(A)$, which will be denoted by ${\alpha}|_{J(A)}$, is also a group-type partial action. Moreover, Lemma \[jsum\] (ii) implies that ${\alpha}|_{J(A)}$ is unital. Applying Theorem \[teo-decomp\] for the partial groupoid action ${\alpha}|_{J(A)}$ we obtain that $J(A)*_\alpha \operatorname{G}\simeq(J(A)\star_{{\beta}}\operatorname{G}_0^2)\star_ \gamma\operatorname{G}(x)$. Consequently $J(A*_\alpha \operatorname{G})\simeq J(A)*_\alpha \operatorname{G},$ as desired.

Right perfect property
----------------------

A ring $R$ is said to be <span>*right perfect*</span> if every right $R$-module has a projective cover. It is known that $R$ is right perfect if and only if $R/J(R)$ is right artinian and $J(R)$ is right $T$-nilpotent (that is, for every infinite sequence of elements of $J(R)$, there is an $n$ such that the product of first $n$ terms is zero). Also, $R$ is right perfect if and only if $R$ satisfies the descending chain condition (DCC) on principal left ideals. These equivalences for the notion of right perfect ring can be seen, for instance, in @AF.

The next result has the same proof of Lemma 3.4 of @P.

\[herej2\] Let $R$ be a ring and $\theta$ a unital global action of $\operatorname{G}$ on $R$. Assume that $I$ is a $\theta$-invariant ideal of $R.$ Then the following assertions hold.

1.  If $I$ is nilpotent, then $I\star_\theta \operatorname{G}$ is nilpotent.

2.  If $I$ is right $T$-nilpotent, then $I\star_\theta \operatorname{G}$ is right $T$-nilpotent.

We need one more auxiliary result.

\[quasi\] Let $\theta=(S_g, \theta_g)$ be a global action of a groupoid $\operatorname{G}$ on a ring $S$ and $I$ a $\theta$-invariant ideal of $S.$ Then $\theta$ induces a global action $\overline{\theta}$ of $\operatorname{G}$ on $S/I$. Moreover, the map $$\Psi:\big(S\star_\theta\operatorname{G}\big)/\big(I\star_\theta\operatorname{G}\big)\to\big(S/I\big)\star_{\overline{\theta}}\operatorname{G},\quad a\delta_g+(I\star_\theta\operatorname{G})\mapsto (a+I)\delta_g, \quad a\in S_{s(g)},$$ is a well-defined ring isomorphism.

Let $g\in \operatorname{G}$. Put $B_g=B_{t(g)}:= \big(S_{t(g)}+I\big)/{I}$. Clearly, $B_g=B_{t(g)}$ is an ideal of $S/I$. As in the proof of Lemma 2.2 of @FL, the ring isomorphism $\theta_g:S_{g{{}^{-1}}}\to S_g$ induces the following ring isomorphism $$\begin{aligned}
        &\overline{\theta}_g:B_{g{{}^{-1}}} \to B_{g},&&\overline{\theta}_g(a+I)=\theta_g(a)+I,&a\in S_{s(g)}.& 
    \end{aligned}$$ Since $I$ is $\theta$-invariant it follows that $\overline{\theta}_g$ is well-defined ring isomorphism, for each $g\in \operatorname{G}$. It is straightforward to check that $\overline{\theta}=(B_g, \overline{\theta}_g)_{g\in \operatorname{G}}$ is a global action of $\operatorname{G}$ on $S/I$. Finally, consider the surjective ring homomorphism $\psi:S\star_\theta\operatorname{G}\to \big(S/I\big)\star_{\overline{\theta}}\operatorname{G}$ given by $\psi(a\delta_g)=(a+I)\delta_g$, for all $a\in S_{s(g)}$. It is clear that $\ker \psi=I\star_\theta\operatorname{G}$ and whence we obtain the ring isomorphism $\Psi$.

The main result of this subsection is the following.

\[rightp\]Suppose that $\alpha$ is finite-type. Then $A\star_{\alpha}\operatorname{G}$ is right perfect if and only if $A$ is right perfect and $\operatorname{G}$ is finite.

Suppose that $A\star_{\alpha}\operatorname{G}$ is right perfect. Observe that by , $A$ is a subring of $A\star_{\alpha}\operatorname{G}$. Moreover $A$ is a right $A$-direct summand of $A\star_{\alpha}\operatorname{G}$. From Proposition 2.1 of @PA follows that $(A\star_{\alpha}\operatorname{G})I\cap A=I$ for every left ideal $I$ of $A$. Since $A\star_{\alpha}\operatorname{G}$ satisfies the (DCC) for left principal ideals follows that $A$ satisfies the (DCC) for left principal ideals. Thus, $A$ is right perfect. Also, by Theorem \[teo-decomp\] we have that $(A\star_{{\beta}}\operatorname{G}_0^2)\star_ \gamma\operatorname{G}(x)$ is right perfect, where $\gamma$ is finite-type (by Lemma \[latoga\]). Then, Theorem 34 of @BC implies that $\operatorname{G}(x)$ is finite. Therefore, by Proposition \[group:connec\] we conclude that $\operatorname{G}$ is finite.

Conversely, suppose that $A$ is right perfect and $\operatorname{G}$ is finite. We claim that $C:=A\star_{{\beta}}\operatorname{G}_0^2$ is right perfect. In fact, fix $B:=A/J(A)$. Notice that $\operatorname{G}_0^2$ acts globally on $A$ by $\beta$ and $J(A)$ is $\beta$-invariant. Hence, by Lemma \[quasi\], $\beta$ induces a global action $\overline \beta$ on $B$. Also, $B$ is right artinian because it is semisimple. As $\operatorname{G}_0^2$ is finite we have by Theorem 1.2 of @NOP that $B*_{\overline\beta}\operatorname{G}_0^2$ is right artinian. Using Lemma \[quasi\] we obtain that $$\label{firstiso}
        B*_{\overline\beta}\operatorname{G}_0^2\simeq C/\big(J(A)\star_{{\beta}}\operatorname{G}_0^2\big) $$ and $C/\big(J(A)\star_{{\beta}}\operatorname{G}_0^2\big)$ is right artinian. We claim that $C/J(C)$ is a ring epimorphic image of $C/\big(J(A)\star_{{\beta}}\operatorname{G}_0^2\big)$. In fact, for $u\in \operatorname{G}_0^2$ and $a_u\delta_u\in J(A)\star_{{\beta}}\operatorname{G}_0^2$ we have $a_u\in J(A)\cap A_{t(u)}$. From Lemma \[jsum\] (v) follows that $a_u1_{C}\in J(C)$. Since $J(C)$ is an ideal of $C$ we obtain that $a_u\delta_u=(a_u1_C)(1_u\delta_u)\in J(C)$ and whence $J(A)\star_{{\beta}}\operatorname{G}_0^2\subset J(C)$. Consequently, the claim is proved and $C/J(C)$ is right artinian. Now we show that $J(C)$ is right $T$-nilpotent. Let $\pi$ be the canonical homomorphism from $C$ onto $C/\big(J(A)\star_{{\beta}}\operatorname{G}_0^2\big)$. Notice that $$\pi(J(C))=J(C)/\big(J(A)\star_{{\beta}}\operatorname{G}_0^2\big)\subseteq  J\Big(C/\big(J(A)\star_{{\beta}}\operatorname{G}_0^2\big)\Big).$$ On the other hand, implies that $J(B*_{\overline\beta}\operatorname{G}_0^2)\simeq J\Big(C/\big(J(A)\star_{{\beta}}\operatorname{G}_0^2\big)\Big),$ consequently $J(C)/\big(J(A)\star_{{\beta}}\operatorname{G}_0^2\big)$ is nilpotent because it is embedded in the nilpotent ring $J(B*_{\overline\beta}\operatorname{G}_0^2)$. Also, using that $A$ is right $T$-nilpotent, it follows from Lemma \[herej2\] (ii) that $J(A)\star_{{\beta}}\operatorname{G}_0^2$ is right $T$-nilpotent. Therefore $J(A\star_{{\beta}}\operatorname{G}_0^2)$ is right $T$-nilpotent. Hence, $C$ is right perfect. Then by Theorem \[teo-decomp\] and Proposition 2.1 of @PA we obtain that $A*_{\alpha} \operatorname{G}$ is right perfect, as desired.

The semiprimary property
------------------------

Recall that a ring $A$ is called <span>*semiprimary*</span> if $A/J(A)$ is right artinian and $J(A)$ is nilpotent. Clearly every semiprimary ring is right perfect.

\[spp\] Suppose that $\alpha$ is finite-type. Then $A\star_{\alpha}\operatorname{G}$ is semiprimary if and only if $A$ is semiprimary and $\operatorname{G}$ is finite.

Assume that $A\star_{\alpha}\operatorname{G}$ is semiprimary. Using Theorem \[teo-decomp\] and Lemma \[latoga\], it follows from Theorem 35 of @BC that $A\star_{{\beta}}\operatorname{G}_0^2$ is semiprimary and $\operatorname{G}(x)$ is finite. Then $\operatorname{G}$ is finite thanks to Proposition \[group:connec\]. Since $A\star_{{\beta}}\operatorname{G}_0^2$ is semiprimary it is right perfect. Hence, Theorem \[rightp\] implies that $A$ is right perfect and whence $A/J(A)$ is right artinian. Also, using (v) of Lemma \[jsum\] we conclude that $J(A)\subseteq J(A\star_{{\beta}}\operatorname{G}_0^2)$. Consequently $J(A)$ is nilpotent.

Conversely, suppose that $A$ is semiprimary and $\operatorname{G}$ is finite. Then $A$ is right perfect. As we saw in the proof of Theorem \[rightp\], $C:=A\star_{{\beta}}\operatorname{G}_0^2$ is right perfect. Thus $C/J(C)$ is right artinian. Since $J(A)$ is nilpotent it follows from Lemma \[herej2\] (i) that $J(A)\star_{{\beta}}\operatorname{G}_0^2$ is nilpotent. Moreover, as in the proof of Theorem \[rightp\], $J(C)/\big(J(A)\star_{{\beta}}\operatorname{G}_0^2\big)$ is nilpotent. Then $J(C)$ is nilpotent and whence $C$ is semiprimary. Finally, since $\operatorname{G}(x)$ is finite, the same argument shows that $C\star_ \gamma\operatorname{G}(x)$ is semiprimary. Thus, the result follows from Theorem \[teo-decomp\].

Krull dimension
---------------

Let $R$ be a unital ring and $M$ a right $R$-module. The Krull dimension of $M$, usually denoted by $\operatorname{Kdim} M_R$ is the deviation of the lattice $\mathcal{L}_R(M)$ of $R$-submodules of $M$. The right Krull dimension of a ring $R$ is defined as $\operatorname{Kdim} R:=\operatorname{Kdim} R_R$. It is well-known that the right modules of Krull dimension $0$ are the nonzero right artinian modules and that every right noetherian module has Krull dimension.

\[krull\] Assume that $\operatorname{G}$ is finite. If $A$ has right Krull dimension then $A\ast_{{\alpha}}\operatorname{G}$ has right Krull dimension and $\operatorname{Kdim} A\ast_{{\alpha}}\operatorname{G}\leq \operatorname{Kdim} A$.

Let $u\in \operatorname{G}_0^2$. Since $A$ has Krull dimension, it follows that $B_u=A_{t(u)}$ has Krull dimension as a right $A$-module. Using that $A_u$ and $A_{t(u)}\delta_{u}$ are isomorphic as right $A$-modules we conclude that $A_{t(u)}\delta_{u}$ has Krull dimension. Since $A\star_{{\beta}}\operatorname{G}_0^2=\bigoplus_{u\in \operatorname{G}_0^2}A_{t(u)}\delta_{u}$, Lemma 6.1.14 of @MR implies that $C:=A\star_{{\beta}}\operatorname{G}_0^2$ has Krull dimension as a right $A$-module. Moreover, it is easy to see that $\operatorname{Kdim} C_A=\operatorname{Kdim}A_A=\operatorname{Kdim}A$. As in the proof of Proposition 41 of @BC, we have that $$\mathcal{L}_{C}(C)\to \mathcal{L}_{A}(C),\quad N\mapsto N,$$ is strictly increasing. Thus, $C$ has Krull dimension and $\operatorname{Kdim} C\leq \operatorname{Kdim} C_A= \operatorname{Kdim}A$. From Theorem \[teo-decomp\] and Proposition 41 of @BC we obtain that $A\star_{\alpha}\operatorname{G}$ has right Krull dimension and $\operatorname{Kdim}A\star_{\alpha}\operatorname{G}\leq \operatorname{Kdim}C\leq  \operatorname{Kdim}A$.

Applications to Leavitt path algebras {#Leavitt}
=====================================

The Leavitt path algebra associated to a directed graph was introduced in @AA and @AAP as follows. Denote a directed graph by $E=(E^0,E^1,r,d)$, where $E^0$ and $E^1$ are countable sets and $r,d : E^1 \to E^0$ are maps. The elements of $E^0$ are called *vertices* and the elements of $E^1$ are called *(real) edges*. A *sink* is a vertex such that no edge emerges out of it. A *path $\mu$ of length n* in $E$ is a sequence of edges $\mu=\mu_1 \cdots \mu_n$ such that $r(\mu_i)=d(\mu_{i+1})$ for $i\in \{1,\ldots,n-1\}$. In such a case, we will write $|\mu|=n,$ $d(\mu):=d(\mu_1)$ is the *source* of $\mu$ and $r(\mu):=r(\mu_n)$ is the *target* of $\mu,$ respectively, a vertex is considered as a path of length $0$. We are using $r,d$, instead $t,s$ to denote the target and source maps of a path because we want to avoid confusion with the notation used for the target and source maps in groupoids. We shall denote $E^{1}(v,\,):=\{ f\in E^1\,:\, d(f)=v \}$.

Let $E=(E^0,E^1,r,d)$ be a directed graph and $\Bbbk$ a field. The *Leavitt path $\Bbbk$-algebra $L_\Bbbk(E)$ of $E$ with coefficients in $\Bbbk$* is the free associative $\Bbbk$-algebra generated by the set $\{v,f,f^{\ast}\, :\, v\in E^0, f\in E^1 \}$ with the following relations:

1.  for all $v,w\in E^0$, $v^2=v$ and $vw=0$ if $v\neq w$,

2.  $d(f)f=fr(f)=f$, for all $f\in E^1$,

3.  $r(f)f^*=f^*d(f)=f^*$, for all $f\in E^1$,

4.  for all $f,f'\in E^1$, $f^*f=r(f)$ and $f^*f'=0$ if $f\neq f'$,

5.  $v=\sum\limits_{ f\in E^{1}(v,\,) } ff^*$, for every $v\in E^0$ such that $d^{-1}(v)$ is non-empty and finite.

The symbols $f^*
$ for $f \in E^1$ are called <span>*ghost edges*</span>, also condition (v) above is called the <span>*Cuntz-Krieger relation*</span>. Notice that $L_\Bbbk(E)=\bigoplus_{v\in E^0}L_\Bbbk(E)v$ and thus $L_\Bbbk(E)$ is unital if and only if $E^0$ is finite. In this case, $1_{L_\Bbbk(E)}=\sum_{v\in E^0}v$.

In what follows in this section, $E=(E^0,E^1,r,d)$ denotes a directed graph and $L_\Bbbk(E)$ is the corresponding Leavitt path algebra, where $\Bbbk$ is a field. Our purpose in this section is to apply the results of the previous section to $L_\Bbbk(E)$.

$L_\Bbbk(E)$ as a partial skew groupoid ring
--------------------------------------------

Leavitt path algebras were realized as partial skew groupoid rings in Theorem 3.11 of @GY. For readers convenience, here we recall this realization.

Firstly, the free path groupoid $\operatorname{G}(E)$ associated to $E$ is constructed in the following way. The set of objects $\operatorname{G}(E)_0$ of $\operatorname{G}(E)$ is $E^0$. We extend the maps $r,d$ from $E^{1}$ to $E^{1}\cup \big(E^{1}\big)^\ast$ by putting: $d(f^*)=r(f)$ and $r(f^*)=d(f)$, for all $f\in E^{1}$. Then, for a finite path $\mu=\mu_1\ldots\mu_n$ in $E$ we put $\mu^{\ast}:=\mu_n^{\ast}\ldots\mu_1^\ast$. Denote by $$P:    =\{\xi_1\xi_2\cdots \xi_n\,:\, n\in {{\mathbb N}}, \,\, \xi_i\in E^1\cup (E^1)^*\cup E^0\,\, \,\text{and}\,\,\, r(\xi_i)=d(\xi_{i+1})\}.$$ An element $\xi_1\ldots\xi_kff^*\xi_{k+1}\ldots\xi_n$ in $P$ can be reduced to $\xi=\xi_1\ldots\xi_k\ldots\xi_n$. Similarly, $\xi$ is the reduction of $$\xi_1\ldots \xi_kd(\xi_{k+1})\xi_{r+1}\ldots\xi_n=\xi_1\ldots\xi_rr(\xi_k)\xi_{r+1}\ldots\xi_n.$$ The elements of $P$ that cannot be reduced are called irreducible. The set of irreducible elements of $P$ will be denoted by $\operatorname{Irr}(P)$ and the reduction of an element $\mu\in P$ will be denoted by $\operatorname{irr}(\mu)$. The set of morphisms of the $\operatorname{G}(E)$ is $\operatorname{Irr}(P)$. Given $\mu,\xi\in \operatorname{Irr}(P)$ such that $r(\mu)=d(\xi)$, the composition is $\mu\cdot\xi=\operatorname{irr}(\mu\xi)$. It is clear that $\xi^{{{}^{-1}}}=\xi^\ast$, for all $\xi\in \operatorname{Irr}(P)$.

\[obs-free\] <span>Notice that in the directed graph we compose from left to right while in the free path groupoid $\operatorname{G}(E)$ we compose from right to left. Thus, <span>*we have that $t(\xi)=d(\xi)$ and $s(\xi)=r(\xi)$, for all $\xi\in \operatorname{G}(E)$*</span>. Precisely, if $\xymatrix{\bullet^{v}\ar@{->}[r]^{\xi}& \bullet^{w}}$ is an edge in $E$ then $d(\xi)=v$ and $r(\xi)=w$. However, $\xi$ as an element of $\operatorname{G}(E)$ has $s(\xi)=w$ and $t(\xi)=v$. Clearly, given $\xi,\eta\in E^{1}$ we have that $\xi\eta$ exists in $E$ if and only if $\eta\xi$ exists in $\operatorname{G}(E)$.</span>

Let $W$ be the set of all finite paths in $E.$ Denote by $W^{\infty}$ the set of all infinite paths in $E$, that is, $\xi\in W^\infty$ if and only if $\xi=\xi_1\xi_2\xi_3\cdots$ with $r(\xi_i)=d(\xi_{i+1})$ and $i\in{{\mathbb N}}$. Also, consider $$\begin{aligned}
    &W^{(1)}=\{a\in W\setminus E^0\,:\,\operatorname{irr}(a)=a \}\quad\text{and} \\[.1em]
    &W^{(2)}=\{(a,b)\in \big(W\setminus E^0\big)^2\,:\,r(a)=r(b)\text{ and }\operatorname{irr}(ab{{}^{-1}})=ab{{}^{-1}}\}.\end{aligned}$$ We define a partial action $\theta$ of $\operatorname{G}(E)$ on the set $$\begin{aligned}
\label{set-X}
    X=\{\xi\in W \,:\, r(\xi) \text{ is a sink}\}\cup  W^{\infty},\end{aligned}$$ in the following way. For each $g\in \operatorname{G}(E)$, let $X_g$ be defined by:

1.  $X_v:=\{\xi\in X\,:\, d(\xi)=v\}$, for all $v\in E^0,$

2.  $X_a:=\{\xi\in X \,:\, \xi_1\xi_2...\xi_{|a|}=a\}$, for all $a\in W^{(1)}$,

3.  $X_{a^{-1}}:=\{\xi\in X \,:\,d(\xi)=r(a)\}$, for all $a\in W^{(1)}$,

4.  $X_{ab^{-1}}:=X_a$, for all $(a,b)\in W^{(2)}$,

5.  $X_g:=\emptyset$, for all other $g \in \operatorname{G}(E)$.

The bijections $\theta_g$, for $g\in \operatorname{G}(E)$, are defined by: $\theta_v:={{\rm id}}_{X_v}$, for all $v\in E^0$, and $$\begin{aligned}
    &\theta_a:X_{a^{-1}}\rightarrow X_a,&  \quad \theta_a(\xi)&=a\xi,\quad a\in \operatorname{G}(E)\setminus E^0& \\[.3em]
    &\theta_{ab^{-1}}:X_{ba^{-1}}\rightarrow X_{ab^{-1}},& \theta_{ab^{-1}}(\xi)&=a\xi_{(|b|+1)}\xi_{(|b|+2)}\ldots,&\end{aligned}$$ for all $(a,b)\in W^{(2)}$. It is clear that $\theta_a$ and $\theta_{ab^{-1}}$ have inverses maps given respectively by: $$\begin{aligned}
    \theta_{a^{-1}}(\xi)&=\begin{cases*}
        r(a),&  \text{if }$ r(a)$ \text{ is  a sink},\\
        \xi_{|a|+1}\xi_{|a|+2}\ldots,& \text{ otherwise},  
    \end{cases*} \\[.3em]
    \theta_{ba^{-1}}(\xi)&=b\xi_{(|a|+1)}\xi_{(|a|+2)}\ldots \end{aligned}$$

By Proposition 3.7 of @GY, the family of pairs $\theta=(X_g, \theta_g)_{g\in\operatorname{G}(E)}$ is a partial action of $\operatorname{G}(E)$ on $X$. Following @GY, denote the algebra of functions from $X$ to $\Bbbk$ by $\mathcal{F}(X)$ and put $\mathcal{F}(X_g):=\{f\in \mathcal{F}(X)\,:\,f\text{ vanishes outside of }X_g\}$ for each $X_{g}\neq \emptyset$ and $\mathcal{F}(X_g):=\{\text{null function}\}$ for each $X_{g}=\emptyset$. Also, define $\tilde{\theta}_g(f)=f\circ \theta_{g^{-1}}$, for all $ f\in \mathcal{F}(X_{g^{-1}})$. It follows from Proposition 3.8 of @GY that $\tilde{\theta}=\big(\mathcal{F}(X_g),\tilde{\theta}_g\big)_{g\in \operatorname{G}(E)}$ is a partial action of $\operatorname{G}(E)$ on $\mathcal{F}(X)$.

For each $g\in \operatorname{G}(E)$, $1_g$ denotes the characteristic of ${X_g}$, that is, $1_g(x)=1$ if $x\in X_g$ and $1_g(x)=0$ if $x\notin X_g$. Notice that $1_g$ is the identity element of $\mathcal{F}(X_g)$. Let $D(X)$ be the $\Bbbk$-subalgebra of $\mathcal{F}(X)$ generated by $\{1_g \,:\, g\in \operatorname{G}(E)\}$. Fix the ideal $D(X)_g:=1_g D(X)$ of $\mathcal{F}(X_g)$, for each $g\in \operatorname{G}$. Observe that $D(X)_g$ is generated (as a subalgebra of $D(X)$) by $\{1_g1_h\,:\, h\in \operatorname{G}(E)\}$. It is clear that the restriction of $\tilde{\theta}_g$ to $D(X)_{g^{-1}}$ is a bijection onto $D(X)_g$. We will denote by $\lambda_g$ the restriction of $\tilde{\theta}_g$ to $D(X)_{g^{-1}}$, that is, $$\begin{aligned}
\label{ac-lam}
    \lambda_g:D(X)_{g^{-1}}\to D(X)_{g}, \qquad  \lambda_g=\tilde{\theta}_g|_{D(X)_{g^{-1}}}.\end{aligned}$$ Then $\lambda=(D(X)_g,\lambda_g)_{g\in \operatorname{G}(E)}$ is a partial action of $\operatorname{G}(E)$ on $D(X)$.

Now we can enunciate Theorem 3.11 of @GY.

\[iso-GY\] $L_{\Bbbk}(E)$ is isomorphic (as a $\Bbbk$-algebra) to $D(X)\star_{\lambda}\operatorname{G}(E)$.

\[rem-cone-case\] <span> Let $E$ be a directed graph. From Proposition 1.2.14 of @AAM follows that $L_{\Bbbk}(E)\simeq \oplus_{i\in I}L_{\Bbbk}(E_i)$, where $E=\sqcup_{i\in I}E_i$ is the decomposition of $E$ in its connected components. For each $E_i$, we denote the set given in by $X_i$, the free path groupoid by $\operatorname{G}(E_i)$ and the action of $\operatorname{G}(E_i)$ on $D(X_i)$ by $\lambda_i$. Thus, Theorem \[iso-GY\] implies that $L_{\Bbbk}(E)\simeq \oplus_{i\in I} D(X_i)\star_{\lambda_i}\operatorname{G}(E_i)$.</span>

\[rem-dist-groups\]<span> The Leavitt path algebra $L_{\Bbbk}(E)$ can be realized as a partial skew group ring cf. Proposition 3.2 of @GR. In this case, the group that acts partially is the free group $\mathbb{F}$ generated by $E^{1}$. On the other hand, suppose that the partial action $\lambda$ given in is group-type, that is, there exist $v\in E^0$ and a transversal ${\tau}(v)$ for $v$ such that $\lambda$ satisfies . Assume that $E^0$ is finite. From Theorem \[teo-decomp\] it follows that $L_{\Bbbk}(E)$ can be realized again as a partial skew group ring. In this case, the group acting partially is the isotropy group $\operatorname{G}(E)(v)$ which in general is easier to deal than $\mathbb{F}$. Hence, sometimes it is more convenient to realize $L_{\Bbbk}(E)$ as a partial skew group ring using the partial action $\lambda$; see Remark \[rem-advantage\].</span>

Determining when $\lambda$ is group-type
----------------------------------------

Let $E$, $D(X)$, $\lambda$ and $\operatorname{G}(E)$ be as in the previous Subsection. In order to apply the results of Section 3 for a Leavitt path algebra $L_{\Bbbk}(E)$, we need to find under what conditions $\lambda$ is a group-type partial action of $\operatorname{G}(E)$ on $D(X)$. Thanks to Remark \[rem-cone-case\], we will assume from now on in this Subsection that $E$ is connected and $E^0$ is finite.

\[decompo\] The following assertions hold.

1.  $D(X)=\bigoplus_{v\in E^0}D(X)_v$.

2.  $X_v\neq \emptyset,$ for all $v\in E^0$.

3.  For every $v\in E^0$ such that $E^1(v,\,)$ is non-empty and finite set, we have that $$1_v=\sum\limits_{ f\in E^1(v,\,)}  1_f\quad \text{and}\quad D(X)_v=\bigoplus \limits_{ f\in E^1(v,\,) }D(X)_f.$$

\(i) We claim that $D(X)=\sum_{v\in E^0}D(X)_v$. In fact, consider an element $a=1_{g_1} 1_{g_2}\cdots 1_{g_n}\in D(X)$ with $g_1, g_2, \dots, g_n\in \operatorname{G}(E)$. Since $1_g1_{t(g)}=1_g$ for all $g\in \operatorname{G}(E)$, we have that $a=1_{t(g_1)}a$. Thus $a\in D(X)_{t(g_1)}\subseteq\sum_{v\in E^0}D(X)_v$. Using that any element of $D(X)$ is a linear combination of elements $a$ as above, we conclude that $D(X)=\sum_{v\in E^0}D(X)_v$. Also, the sum is direct because $D(X)_v=D(X)1_v$ and $1_{v_i}1_{v_j}=\delta_{i,j}1_{v_i},$ for all $v,v_i,v_j\in E^0$.

\(ii) Let $v\in E^0$. If $v$ is a sink then $v\in X_v$ and $X_v$ is non-empty. Suppose that $v$ is not a sink. In this case, there exists $\xi_1\in E^1 $ such that $d(\xi_1)=v$. If $r(\xi_1)=v_1$ is a sink then $\xi_1\in X_v$. Otherwise, there exists $\xi_2\in E^1 $ such that $d(\xi_2)=v_1$. If $r(\xi_2)=v_2$ is a sink then $\xi_1\xi_2\in X_v$. Otherwise, we repeat the process. If this process is finite then we obtain an element $\xi_1\xi_2\cdots \xi_n \in X_v$. If the process is infinite then we have an infinite path $\xi=\xi_1\xi_2\cdots $ for which $d(\xi)=v$, that is, $\xi\in X_v$.

\(iii) Let $v\in E^0$ such that $E^1(v,\,)$ is non-empty and finite. Given $f,f'\in E^1$, with $f\neq f'$ and $f,f'\in d^{-1}(v)$, we have that $X_{f}\cap X_{f'}=\emptyset$. Hence, in order to show the equality on the left side in (iii) it is enough to verify that $X_v=\bigcup\limits_{ f\in E^1(v,\,) } X_f$. The inclusion $\supseteq$ is immediate. For the reverse, consider $\xi\in X_v$. Then there are $f\in E^1$ and $\eta\in W\cup W^{\infty}$ with $r(f)=d(\eta)$ such that $\xi=f\eta$. Then $\xi\in X_f$ and $d(f)=d(\xi)=v$ which implies the inclusion $\subseteq$. The equality $D(X)_v=\bigoplus \limits_{ f\in E^1(v,\,)}D(X)_f$ follows from the fact that $1_f1_p=\delta_{f,p}1_f$, for all $f,p\in E^1.$

For the next result, we recall that if $\xi$ is an edge in $E$ such that $d(\xi)=v$ and $r(\xi)=w$ then $\xi$ as an element of $\operatorname{G}(E)$ has $s(\xi)=w$ and $t(\xi)=v$; see Remark \[obs-free\]. We also denote $W^{(-1)}:=\{a{{}^{-1}}\,:\,a\in W^{(1)}\}$.

\[gtype\] Suppose that $E$ is connected. Then $\lambda$ is a group-type partial action of $\operatorname{G}(E)$ on $D(X)$ if and only if there are a vertex $v\in E^0$ of $E$ and a transversal $\tau(v)=\{\tau_{w}:v\to w\,:\, w\in E^0\}$ for $v$ in $G(E)$ such that $\tau_w\in W^{(-1)}\cup W^{(1)}\cup W^{(2)}$ for all $w\in E^0, w\neq v$ and

1.  if $\tau_w\in W^{(-1)}$ then

    $$\begin{aligned}
                \{\xi\in X: \xi_1\xi_2...\xi_{|\tau{{}^{-1}}_w|}=\tau{{}^{-1}}_w\}=\{\xi\in X : d(\xi)=v\}, \label{eqq1-tg} 
            \end{aligned}$$

2.  if $\tau_w\in W^{(1)}$ then

    $$\begin{aligned}
                \{\xi\in X: \xi_1\xi_2...\xi_{|\tau_w|}=\tau_w\}=\{\xi\in X : d(\xi)=w\}, \label{eq1-tg} 
            \end{aligned}$$

3.  if $\tau_w=a_wb^{-1}_w$ with $(a_w,b_w)\in W^{(2)}$ then

    $$\begin{aligned}
                \{\xi\in X: \xi_1\xi_2...\xi_{|b_w|}=b_w\}=\{\xi\in X : d(\xi)=v\}  , \label{eq2-tg}
            \end{aligned}$$

4.  if $\tau_w=a_wb^{-1}_w$ with $(a_w,b_w)\in W^{(2)}$ then

    $$\begin{aligned}
                \{\xi\in X: \xi_1\xi_2...\xi_{|a_w|}=a_w\}&=\{\xi\in X : d(\xi)=w\}, \label{eq3-tg}
            \end{aligned}$$

Notice that $\lambda$ is a group-type partial action of $\operatorname{G}(E)$ on $D(X)$ if and only if there are $v\in E^0$ and a transversal $\tau(v)=\{\tau_{w}: v\to w \,:\, w\in \operatorname{G}(E)_0=E^0\}$ for $v$ such that $D(X)_{\tau{{}^{-1}}_w}=D(X)_v$ and $D(X)_{\tau_w}=D(X)_w$, for all $w\in E^0$. Consequently, $\lambda$ is a group-type if and only if $1_{\tau{{}^{-1}}_w}=1_v$ and $1_{\tau_w}=1_w$, for all $w\in E^0$, which is equivalent to $$\begin{aligned}
\label{xeq}X_{\tau{{}^{-1}}_w}=X_v=\{\xi\in X \,:\, d(\xi)=v\},\quad X_{\tau_w}=X_w=\{\xi\in X \,:\, d(\xi)=w\},
    \end{aligned}$$ for all $w\in E^0$. Using the definition of the ideals $X_g$, $g\in \operatorname{G}(E)$, given in the previous subsection and (ii) of Lemma \[decompo\], we obtain directly from that $\lambda$ is group-type if and only if the transversal ${\tau}(v)$ satisfies , , and .

\[ssink\] <span>Notice that the inclusions $\subset $ in equations , , and always hold. Also, suppose that $v\in E^0$ is a sink and that there exists a transversal $\tau(v)=\{\tau_{w}:v\to w\,:\, w\in E^0\}$ in $\operatorname{G}(E)$ for $v$ contained in $W$. In this case ${\tau}(v)\setminus\{v\}\subset   W^{(1)}$ and $\lambda$ is a group-type partial action of $\operatorname{G}(E)$ on $D(X)$ with respect to the transversal $\tau(v)$ if and only if holds.</span>

For the reader's convenience we recall that a path $\xi=\xi_1\xi_2\ldots \xi_n$ in $E$ is <span>*closed*</span> if $d(\xi)=r(\xi)$. If $\xi$ is closed and $d(\xi_i)\neq d(\xi_j)$, for every $i\neq j$, then $\xi$ is called <span>*a cycle*</span>. For a cycle $\xi$ in $E$, $\xi^{\infty}$ denotes the infinity path $\xi\xi\ldots$ in $E$. A directed graph is said <span>*acyclic*</span> if it has no cycle.

The next result gives necessary and sufficient conditions on $E$ for $\lambda$ to be group-type in the case that $E$ has a sink and $|E^{0}|\geq 2$.

\[daniel\] Suppose that $|E^{0}|\geq 2$ and that $E$ has a sink $v.$ Then $\lambda$ is a group-type partial action of $\operatorname{G}(E)$ on $D(X)$ if and only if $|d^{-1}(w)|= 1$, for all $w\in E^{0}\setminus\{v\}$ and $E$ is acyclic.

Let $v\in E^{0}$ a sink. Since $\lambda$ is group-type, by (i) of Remark \[obs-not-depend\] and Theorem , we can take a transversal $\tau(v)=\{{\tau}_{w}\}_{w\in E^0}$ in $\operatorname{G}(E)$ for $v$ such that $\tau_w\in W^{(-1)}\cup W^{(1)}\cup W^{(2)}$ for all $w\in E^0, w\neq v$, and satisfies , , and . Let $w\in E^0\setminus\{v\}$. If $\tau_w\in W^{(-1)}$ then ${\tau}{{}^{-1}}_w\in W$ and $t({\tau}{{}^{-1}}_w)=v$. Hence, there exists $\xi\in E^{1}$ such that $d(\xi)=v$ which is a contradiction because $v$ is a sink. If ${\tau}_w=a_wb{{}^{-1}}_w$ with $(a_w,b_w)\in W^{(2)}$ then $t(b_w)=v$. Again, there is $\xi\in E^{1}$ such that $d(\xi)=v$ which is an absurd. Hence, ${\tau}_w\in W^{(1)}$. In particular $d(\tau_w)=w$ and $|d^{-1}(w)|\geq 1$. Suppose that there exists $w_0\in E^{0}$ with $|d^{-1}(w_0)|>1$. If ${\tau}_{w_0}=\xi_1\ldots\xi_r\in W^{(1)}$ then there is $\eta_1\in E^1$ with $\eta_1\neq \xi_1$ and $d(\eta_1)=w_0$. If $r(\eta_1)$ is a sink then $\eta_1\in X$. Otherwise, there is $\eta_2\in E^1$ with $d(\eta_2)=r(\eta_1)$. If $r(\eta_2)$ is a sink then $\eta_1\eta_2\in X$. Otherwise, we repeat the process. If the process is finite then we obtain an element $\eta=\eta_1\ldots\eta_t\in X$ such that $d(\eta)=w_0$. If the process is infinite, then there is an infinite path $\eta=\eta_1\eta_2\ldots\in X$ such that $d(\eta)=w_0$. In both cases, we have that $\eta\in \{\xi\in X : d(\xi)=w_0\} \setminus \{\xi\in X: \xi_1\xi_2...\xi_{|\tau_{w_0}|}=\tau_{w_0}\}$ which implies that does not hold and we have a contradiction. Now suppose that $\nu=\nu_1\nu_2\ldots \nu_n$ is a cycle in $E$. Thus $\nu^\infty\in X$. If $\nu^\infty=\tau_{d(\nu_1)}\eta$ for some path $\eta$ in $E$ then $v=r(\tau_{d(\nu_1)})=d(\eta),$ which is a contradiction. Hence $\nu^\infty\in \{\xi\in X : d(\xi)=d(\nu_1)\} \setminus \{\xi\in X: \xi_1\xi_2...\xi_{|\tau_{d(\nu_1)}|}=\tau_{d(\nu_1)}\}$ and again we have a contradiction. So $E$ is acyclic.

Conversely, since $E^0$ is finite, $E$ is acyclic and $|d^{-1}(w)|= 1$, for every $w\in E^{0}\setminus v$, it follows that there are no infinite paths in $E$. Also, for any $w\in E^0\setminus\{v\}$, there is a unique (finite) path $\tau_w$ in $E$ with $d(\tau_w)=w$ and $r(\tau_w)=v$. Indeed, write $w=w_1$ and let $\eta_1\in E^1$ with $d(\eta_1)=w_1$. As $E$ is acyclic we have $w_2=r(\eta_1)\neq w_1$. If $w_2=v$ we take $\tau_w=\eta_1$ or else we repeat the process. Since $E^0$ is finite, the process is also finite. Thus, there are vertices $w_1=w,\ldots,w_n=v$ with $w_i\neq w_j$ for all $1\leq i\neq j\leq n$ and edges $\eta_1,\ldots,\eta_n$ with $d(\eta_i)=w_i$ and $r(\eta_i)=w_{i+1}$, for all $1\leq i\leq n-1$. Hence $\tau_w=\eta_1\cdots \eta_n$ satisfies $d({\tau}_w)=w$ and $r({\tau}_w)=v$. Suppose that $\xi$ is a path in $E$ such that $d(\xi)=w$ and $r(\xi)=v$. Using that $E$ has no infinite paths, we conclude that $\xi=\xi_1\ldots\xi_r$. Since $|d^{-1}(w)|= 1$, we have that $\eta_1=\xi_1$. Hence, $d(\eta_2)=r(\eta_1)=r(\xi_1)=d(\xi_2)$ which implies $\xi_2=\eta_2$. In this way one obtains $r=n$ and $\eta_i=\xi_i$, for all $1\leq i\leq n$. Therefore $\xi=\tau_w$. Finally, notice that $\{\mu\in X: \mu_1\mu_2...\mu_{|\tau_w|}=\mu_w\}=\{\mu\in X : d(\mu)=w\}=\{{\tau}_w\}$ and whence holds. Thus, $\lambda$ is a group-type partial action of $\operatorname{G}(E)$ on $D(X)$ with transversal $\tau(v)=\{{\tau}_w\}_{w\in E^0}$ for $v$.

\[cor-matrices\] Suppose that $|E^{0}|\geq 2$ and that $E$ has a sink $v$. If $\lambda$ is a group-type partial action of $\operatorname{G}(E)$ on $D(X)$ then $$L_{\Bbbk}(E)\simeq D(X)\star_{\beta} (E^0)^2\simeq {\rm M}_n(\Bbbk).$$

Let $v$ be the unique sink of $E$. By (i) of Remark \[obs-not-depend\] and Theorem , there exists a transversal $\tau(v)=\{{\tau}_{w_j}\,:\,1\leq j\leq n\}$ in $\operatorname{G}(E)$ for $v$ that satisfies , , and . Since $\operatorname{G}(E)(v)$ is the trivial group $\{v\}$, it follows from Theorem \[teo-decomp\] and Theorem \[iso-GY\] that $L_{\Bbbk}(E)$ is isomorphic to $D(X)\star_{\beta} (E^0)^2$, where $\beta=(B_u,\beta_u)_{u\in (E^0)^2}$ is the global action of $\operatorname{G}(E)_0^2=(E^0)^2$ on $D(X)$ given by . Explicitly, if $u=(w_i,w_j)\in (E^0)^2$ then $B_u=D(X)_{w_j}=D(X)1_{w_j}$. As we saw in the proof of Proposition \[daniel\], $X_{{\tau}_{w}}=\{{\tau}_w\}$, for all $w\in E^{0}$. Thus $1_{w_j}1_g=0$ if $\tau_{w_j}\notin X_g$ or $1_{w_j}1_g=1_{w_j}$ if $\tau_{w_j}\in X_g$, for all $g\in \operatorname{G}(E)$. Hence $D(X)1_{w_j}=\Bbbk\langle 1_{w_j}\rangle$ which is isomorphic (as algebra) to $\Bbbk$. Also, notice that $\beta_u:\Bbbk\langle 1_{w_i}\rangle \to \Bbbk\langle 1_{w_j}\rangle$ is given by $\beta_u(a1_{w_i})=a1_{w_j}$, for all $a\in \Bbbk$. Hence it is clear that $$\begin{aligned}
        \psi: D(X)\star_{\beta} (E^0)^2\to{\rm M}_n(\Bbbk),\quad \psi(a1_{w_j}\delta_{(w_i,w_j)})=ae_{ji},\quad a\in\Bbbk,
    \end{aligned}$$ is an algebra isomorphism, where $e_{ji}$ is the elementary matrix that has $1$ in the $(j,i)$-entry and $0$ in the other entries.

<span> The result of the previous corollary is known; see, for instance, Theorem 2.6.17 of @AAM. </span>

\[toe\] <span>Consider the *Toeplitz graph* $E_T$ given by $$\begin{aligned}
            \xymatrix{ &\bullet^{u}\ar@(dl,dr)_{\xi_1} \ar@{->}[r]^{\xi_2}&\bullet^{v}} 
        \end{aligned}$$ Then follows by Proposition \[daniel\] that $\lambda$ is not a group-type partial action of $\operatorname{G}(E_T)$ on $D(X)$.</span>

Notice that Proposition \[daniel\] is not true if $E$ has no sink. Indeed, in the next example, $\lambda$ is a group-type partial action of $\operatorname{G}(E)$ on $D(X)$ and $|d^{-1}(u)|=2,$ for some vertex $u$.

\[contra-exemplo\] <span> Consider the directed graph $E$ given by $$\begin{aligned}
            \xymatrix{ &                                  &\hspace*{.4cm}\bullet^{w_1}\ar@{->}[rd]^{\xi_1}&  &  \\  
                &\bullet^{u}\ar@{->}[ru]^{\xi_3}\ar@{->}[rd]_{\xi_4} & &\bullet^{v} \ar@{->}[ll]_{\eta}& \\
                &                                                    &\hspace*{.4cm}\bullet_{w_2} \ar@{->}[ru]_{\xi_2} &                      &   }
        \end{aligned}$$ Consider the set $\tau(v)=\{{\tau}_v=v,\,{\tau}_{w_1}=\xi_1,\,{\tau}_u=\eta^{-1},\,\tau_{w_2}=\xi_2\}$. It is clear that ${\tau}_{w_1}$ and ${\tau}_{w_2}$ satisfy while ${\tau}_u$ verifies . Hence, $\lambda$ is a group-type partial action of $\operatorname{G}(E)$ on $D(X)$ and $|d^{-1}(u)|=2$.</span>

Recall that $E$ is <span>*finite*</span> if both $E^0$ and $E^1$ are finite sets.

\[conse\]<span>If $E$ is finite and $E$ has no sinks, then $E$ contains a cycle. Thus $L_{\Bbbk}(E)$ is neither semisimple (Corollary 4.2.13 of @AAM) nor von Neumann regular (Theorem 3.4.1 of @AAM).</span>

\[LPA0\]<span>Let $n\in {{\mathbb N}}$, $n>1$ and consider the directed graph $E=A_n$: $$\begin{aligned}
            \xymatrix{\bullet^{v_1}\ar@{->}[r]^{\xi_1}& \bullet^{v_2}\ar@{->}[r]^{\xi_2}& \bullet^{v_3}\ar@{.}[r]&\bullet^{v_{n-1}}\ar@{->}[r]^{\xi_{n-1}}&\bullet^{v}.}    
        \end{aligned}$$ By Corollary \[cor-matrices\], we have that $L_{\Bbbk}(E)\simeq D(X)\star_{\beta} (E^0)^2\simeq{\rm M}_n(\Bbbk)$.</span>

\[LPA1\]

In the previous example, we take a transversal of a vertex $v$ which is a sink. Now we present an example where the vertex is not a sink. Consider the directed graph $E$:

$$\begin{aligned}
            \xymatrix{ &\bullet^{v_2} \ar@{->}[r]^{\xi_2}&\bullet^{v_1}\ar@{->}[rd]_{\xi_1}&\bullet^{v_3}\hspace{-0.35cm}\ar@{->}[d]^{\xi_3}& \bullet^{v_4}\ar@{->}[ld]^{\xi_4} \\
                &                                 &                               &\hspace{0.35cm}\bullet^{v_5}\ar@(dl,dr)_{\xi_5}                & &} 
        \end{aligned}$$

Clearly, $\operatorname{G}(E)$ is connected. Let $\tau(v_1)=\{\tau_{v_i}:v_1\to v_i\}_{1\leq i\leq 5}$ be a transversal for $v_1$ given by $$\begin{aligned}
            &\tau_{v_1}={v_1},& &\tau_{v_2}=\xi_2,& &\tau_{v_i}=\xi_{i}\xi{{}^{-1}}_1,\,\,\, \text{for all } 3\leq i\leq 5.&   
        \end{aligned}$$ It is directly to verify that ${\tau}(v_1)\setminus\{v_1\}\subset W^{(1)}\cup W^{(2)}$ and , and are satisfied. Then, by Proposition \[gtype\], $\lambda$ is a group-type partial action of $\operatorname{G}(E)$ on $D(X)$. Notice that $\operatorname{G}(E)(v_1)$ is the infinite cyclic group generated by $\xi_1\xi_5\xi_1{{}^{-1}}$ which is isomorphic (as a group) to ${{\mathbb Z}}$. By Theorem \[teo-decomp\], $L_{\Bbbk}(E)\simeq C\star_{\gamma}{{\mathbb Z}}$, where $C=D(X)\star_{\beta}(E^0)^2$. We will calculate $C$ explicitly. Denote by $\xi^{\infty}_5$ the infinite path $\xi_5\xi_5\xi_5\cdots $. Then $X_{v_i}=\{\xi_i\xi^{\infty}_5\}$ for all $i\neq 2$ and $X_{v_2}=\{\xi_2\xi_1\xi^{\infty}_5\}$. Hence, as in Example \[LPA0\], we get $C\simeq {\rm M}_5(\Bbbk)  $ and $L_{\Bbbk}(E)\simeq {\rm M}_5(\Bbbk)\star_{\gamma}{{\mathbb Z}}$.

<span>Let $n$ be a positive integer and $R_n$ the *rose with n petals*, that is, $R_n$ is the directed graph that has one vertex $v$ and $n$ loops $\xi_i$, with $1\leq i\leq n$. Here $\operatorname{G}(E)=\operatorname{G}(E)(v)$ is a group. Thus the only transversal possible to $v$ is $\tau(v)=\{v\}$. In this case, the action $\beta$ of $\operatorname{G}(E)$ on $D(X)$ is trivial and $L_{\Bbbk}(R_n) \simeq D(X)\star_{\gamma}G(E)$. Moreover, $X=X_v=X_{\xi_i{{}^{-1}}}=W^\infty$ and, by Lemma \[decompo\], $D(X)=\bigoplus_{i=1}^nD(X)_{\xi_i}$. It is straightforward to check that $$\begin{aligned}
\label{eq-ex}
            D(X)_{\xi{{}^{-1}}_j}=D(X),\quad \lambda_{\xi_j}(1_{D(X)})=1_{\xi_j}\,\,\text{ and }\,\, \lambda_{\xi{{}^{-1}}_j}(1_{\xi_i})=\delta_{i,j}1_{D(X)}.            
        \end{aligned}$$ Since $G(E)_0=E^0=\{v\}$, it is easy to verify that the partial action $\gamma$ of $\operatorname{G}(E)_v$ on $D(X)$ given in Lemma 4.3 of @BPP is equal to $\lambda$. Thus $L_{\Bbbk}(R_n) \simeq D(X)\star_{\lambda}G(E)$. On the other hand, let $L_{\Bbbk}(1,n)$ be the free associative $\Bbbk$-algebra with $2n$ generators $\{x_1,\ldots,x_n,y_1,\ldots,y_n\}$ subject to the relations $y_jx_i=\delta_{i,j}1_R$ and $\sum_{i=1}^{n}x_iy_i=1_R$. Consider the linear map $\varphi$ from the free associative algebra $\Bbbk\langle x_1,\ldots,x_n,y_1,\ldots,y_n\rangle$ onto $D(X)\star_{\lambda} \operatorname{G}(E)$ given by $$\begin{aligned}
            &\varphi(y_j)=1_{D(X)}\delta_{\xi{{}^{-1}}_j}, & &\varphi(x_i)=1_{\xi_i}\delta_{\xi_i}.
        \end{aligned}$$ By , we have that $$\begin{aligned}
            \varphi(y_j)\varphi(x_i)&=\big(1_{D(X)}\delta_{\xi{{}^{-1}}_j}\big) \big(1_{\xi_i}\delta_{\xi_i}\big)=\delta_{i,j}1_{D(X)}\delta_{\xi{{}^{-1}}_j\xi_i}=\delta_{i,j}1_{D(X)\star_{\lambda} \operatorname{G}(E)},\\
            \sum_{i=1}^{n}\varphi(x_i\varphi(y_i)&=\sum_{i=1}^{n}\big(1_{\xi_i}\delta_{\xi_i}\big)\big(1_{D(X)}\delta_{\xi{{}^{-1}}_i}\big)=\sum_{i=1}^{n}1_{\xi_i}\delta_{1}= 1_{D(X)\star_{\lambda} \operatorname{G}(E)}.
        \end{aligned}$$ Hence, $\varphi$ induces an algebra homomorphism $\psi:L_{\Bbbk}(1,n)\to D(X)\star_{\lambda} \operatorname{G}(E)$ which is bijective. Then $L_{\Bbbk}(R_n) \simeq L_{\Bbbk}(1,n)$ and we recover Proposition 1.3.2 of @AAM. </span>

Ring theoretic properties of $L_{\Bbbk}(E)$
-------------------------------------------

Given a directed graph $E,$ we apply the results of Section 3 to relate algebraic properties of the ring $D(X)$ to algebraic properties of the Leavitt path algebra $L_{\Bbbk}(E).$

### Noetherianity of $L_{\Bbbk}(E)$

We shall relate noetherian properties of $L_{\Bbbk}(E)$ and $D(X).$ We start with the following.

\[noetd\] The ring $D(X)$ is noetherian if and only if $E^0$ is finite and $D(X)_v$ is a noetherian ring, for all $v\in E^0$.

Assume that $D(X)$ is noetherian and suppose that $E^0=\{v_0,v_1, \ldots\}$ is infinite. Consider, for each $j\geq 0$, the left ideal $I_j:=D(X)_{v_0}\oplus\ldots\oplus D(X)_{v_j}$ of $D(X)$. Since $1_{v_i}1_{v_j}=\delta_{ij}1_{v_i}$, we have that $1_{v_{j+1}}\in I_{j+1}\setminus I_j$. Hence, $I_j\subsetneq I_{j+1}$ for all $j\geq 0$. Thus we have a contradiction because $D(X)$ is noetherian. Now consider $v\in E^0$ and $I$ a left ideal of $D(X)_v$. By Lemma \[decompo\], $I$ is a left ideal of $D(X)$ and whence $I$ is finitely generated as left $D(X)$-module. Then $I$ is finitely generated as left $D(X)_v$-module because $D(X)_v$ is a direct summand of $D(X)$. The converse is immediate from well-known results of noetherian rings.

<span>*From now on we shall assume that $\operatorname{G}(E)$ is connected, $E^0$ is finite and that there are a vertex $v\in E^0$ and a transversal $\tau(v)=\{\tau_{w}:v\to w\,:\, w\in E^0\}$ for $v$ such that ${\tau}(v)\setminus\{v\}\subset W^{(-1)}\cup W^{(1)}\cup W^{(2)}$ and , , and are satisfied*</span>.

Observe that as $\lambda$ is a group-type partial action of $\operatorname{G}(E)$ on $D(X)$ we obtain from Theorem \[teo-decomp\] that $L_{\Bbbk}(E)\simeq D(X)\star_{\lambda}\operatorname{G}(E)$. Also, if $\operatorname{G}(E)(v)$ is finite then $\operatorname{G}(E)$ is finite by Proposition \[group:connec\].

For the reader's convenience, we recall some extra notions on a directed graph $E$. Let $\xi=\xi_1\xi_2\ldots \xi_n$ be a path in $E$. An element $\eta\in E^1$ is called an <span>*exit for $\xi$*</span> if there exists $1\leq i\leq n$ such that $d(\eta)=d(\xi_i)$ and $\eta\neq \xi_l$ and we say that $E$ satisfies <span>*Condition (NE)*</span> if there is no cycle in $E$ with an exit.

\[noethercond\] The following assertions are valid.

1.  If $L_{\Bbbk}(E)$ is left noetherian then $D(X)$ and $D(X)\star_{{\beta}} (E^0)^2$ are left noetherian.

2.  Suppose that $G(E)(v)$ is a polycyclic-by-finite group. The following statements are equivalent:

    1.  $L_{\Bbbk}(E)$ is left noetherian,

    2.  $D(X)$ is left noetherian,

    3.  $D(X)_v$ is left noetherian, for all $v\in E^0$,

    4.  $E$ is finite and it satisfies Condition (NE).

Item (i) follows directly from Theorem \[noete\] (i). The equivalence (a) $\Leftrightarrow$ (d) in (ii) follows from Theorem 3.10 of @AAM2 and (b) $\Leftrightarrow$ (c) is Lemma \[noetd\]. From Theorem \[noete\] (ii) we obtain that (b) $\Rightarrow$ (a). Notice that (a) $\Rightarrow$ (b) by item (i).

\[obs-true-general\] <span> The equivalence between (a) and (d) in (ii) of Proposition \[noethercond\] was proved for general directed graphs in Theorem 3.10 of @AAM2. </span>

\[rem-advantage\] <span>Since ${{\mathbb Z}}$ is a polycyclic-by-finite and ${\rm M}_5(\Bbbk)$ is left noetherian it follows from Proposition \[noethercond\] (ii) that the algebra $L_{\Bbbk}(E)$ from Example \[LPA1\] is left noetherian. If we realize $L_{\Bbbk}(E)$ as a partial skew group ring as in Proposition 3.2 of @GR, we obtain that $L_{\Bbbk}(E)$ is isomorphic to the partial skew group ring $D_0\star_{\theta}\mathbb{F}_5$, where $\mathbb{F}_5$ is the free group generated by $E^{1}$, $D_0$ is a subalgebra of the algebra of functions on a set $X$ and $\theta$ is a partial action of $\mathbb{F}_5$ on $D_0$. It was proved in @CCF that a partial skew group ring of a noetherian ring by a partial action of a polycyclic-by-finite group is noetherian. It is well-known that $\mathbb{F}_5$ is not polycyclic-by-finite. Hence we can not use @CCF for to conclude that $L_{\Bbbk}(E)$ is noetherian. </span>

### Regularity of $L_{\Bbbk}(E)$

There are several results in the literature on group graded von Neumann regularity for Leavitt path algebras; see, for instance, Theorems 1.1, 1.3 and 1.4 in @La2. In particular, $L_{\Bbbk}(E)$ with the standard ${{\mathbb Z}}$-grading is graded von Neumann regular. Our next result relates groupoid graded von Neumann regularity of $L_{\Bbbk}(E)$ with von Neumann regularity of the rings $D(X)$ and $L_{\Bbbk}(E)$ itself.

Suppose that $\operatorname{ch}(\Bbbk)$ does not divide $\vert E^0 \vert1_{\Bbbk}$.

1.  The following statements are equivalent:

    1.  $D(X)$ is von Neumann regular,

    2.  $D(X)\star_{{\beta}}\operatorname{G}_0^2$ is graded von Neumann regular,

    3.  $D(X)\star_{{\beta}}\operatorname{G}_0^2$ is von Neumann regular,

    4.  $L_{\Bbbk}(E)$ is $\operatorname{G}(E)$-graded von Neumann regular.

2.  If $\operatorname{G}(E)(v)$ is finite and $ {\rm tr}_{\lambda_{\operatorname{G}(E)(v)}}(1_v)$ is invertible in $D(X)_v$ then:

    1.  The ring $D(X)$ is semiprimitive, that is $J(D(X))=\{0\}$.

    2.  If $L_{\Bbbk}(E)$ is $\operatorname{G}(E)$-graded von Neumann regular then it is von Neumann regular,

Firstly, notice that $\vert E^0 \vert 1_{\Bbbk}$ is invertible in $D(X)$. Also, by Theorem \[teo-decomp\], we have that $L_{\Bbbk}(E)\simeq D(X)*_{\lambda} \operatorname{G}(E)$ because $\lambda$ is a group-type partial action of $\operatorname{G}(E)$ on $D(X)$. Thus (i) follows from Theorem \[vn1\].

\(ii) From Theorem \[jaco\], $J(L_{\Bbbk}(E))\simeq J(D(X))*_\lambda \operatorname{G}(E)$. However, by Proposition 6.3 of @AAP, $J(L_{\Bbbk}(E))=\{0\}$ and whence $J(D(X))=\{0\}$. Hence $D(X)$ is semiprimitive. For (b), observe that from the equivalence (a) $\Leftrightarrow$ (d) in part (i) we obtain that $D(X)$ is von Neumann regular. Hence (b) follows by (e) in Theorem \[vn1\].

<span>Assume that $\operatorname{ch}(\Bbbk)\neq 5$ and consider the Leavitt path algebra given in Example \[LPA1\], that is, $L_{\Bbbk}(E)\simeq {\rm M}_5(\Bbbk)\star_{\gamma}{{\mathbb Z}}$. Since ${\rm M}_5(\Bbbk)$ is von Neummann regular, it follows from Proposition 3.10 of @La2 and Remark \[obs-g-grade\] that $L_{\Bbbk}(E)$ is $\operatorname{G}(E)$-graded von Neumann regular. Hence, the equivalences in (i) of the previous proposition are satisfied. Also, Remark \[obs-g-grade\] implies that $L_{\Bbbk}(E)$ is $\operatorname{G}(x)$-graded von Neumann. However, by Theorem 3.4.1 of @AAM, $L_{\Bbbk}(E)$ is not von Neumann regular because $E$ has one cycle.</span>

<span>We have by (i) of Lemma \[decompo\] that $D(X)$ is a direct sum of orthogonal idempotents rings. Thus, by Lemma \[vnr\], $D(X)$ is von Neumann regular if and only if $D(X)_v$ is von Neumann regular, for all $v\in E^0$. </span>

### Right perfect, semiprimary and Krull dimension

We end the paper with the following remark about other ring theoretic properties of $L_{\Bbbk}(E)$.

1.  $L_{\Bbbk}(E)$ is right perfect if and only if $D(X)$ is right perfect and $\operatorname{G}(E)(v)$ is finite; this follows from Theorem \[rightp\].

2.  $L_{\Bbbk}(E)$ is semiprimary if and only if $D(X)$ is semiprimary and $\operatorname{G}(E)(v)$ is finite; this follows from Theorem \[spp\]

3.  If $\operatorname{G}(E)(v)$ is finite and $D(X)$ has right Krull dimension then $L_{\Bbbk}(E)$ has right Krull dimension and $\operatorname{Kdim} L_{\Bbbk}(E)\leq \operatorname{Kdim} D(X)$. In fact, this is an immediate consequence of Theorem \[krull\] since $\operatorname{G}(E)$ is finite.

<span>99</span> G. Abrams and G. Aranda-Pino, The Leavitt path algebra of a graph, <span>*J. Algebra*</span> **293**, 319–334 (2005)

G. Abrams and G. Aranda-Pino, The Leavitt path algebras of arbitrary graphs, <span>*Houston J. Math.*</span> **34** (2), 423–442 (2008)

G. Abrams, P. Ara and M. S. Molina, Locally finite Leavitt path algebras. <span>*Israel J. Math.*</span> **165**(1), 329–348 (2008)

G. Abrams, P. Ara and M. S. Molina, <span>*Leavitt Path Algebras*</span>, Lecture Notes in Mathematics 2191, Springer-Verlag (2017)

F. Anderson and K. R. Fuller, <span>*Rings and categories of modules*</span>, Springer-Verlag, Berlin, Heidelberg and New York (1974)

D. Bagio, D. Flôres and A. Paques, Partial actions of ordered groupoids on rings, <span>*J. Algebra Appl.*</span> **9**, 501–517 (2010)

D. Bagio and A. Paques, Partial Groupoid Actions: Globalization, Morita theory, and Galois theory, *Comm. Algebra* **40**, 3658–3678 (2012)

D. Bagio, A. Paques, and H. Pinedo, On partial skew groupoid rings, *Internat. J. Algebra Comput.* (to appear).

D. Bagio, A. Paques, and H. Pinedo, Restriction and extension of partial actions. *J. Pure Appl. Algebra* **224** (10), 106391 (2020),

D. Bagio and H. Pinedo, On the separability of the partial skew groupoid ring. *São Paulo J. Math. Sci.* **11**, 370–384 (2017)

L. Bemm. and W. Cortes, Ring theoretic properties of partial crossed products and related themes, <span>*J. Algebra*</span> **493**, 267-296 (2018)

P. A. A. B. Carvalho, W. Cortes, M. Ferrero, Partial skew group rings over polycyclic by finite groups, <span>*Algebr. Represent. Theory*</span> **14**, 449–-462 (2009)

M. Ferrero, J. Lazzarin, Partial actions and partial skew group rings, <span>*J. Algebra*</span> **319**, 5247–5264 (2008)

D. Flôres, A. Morgado and A. Paques, Groupoid gradings: A Morita context, semiprimality, and a Bergman-type question, <span>*Comm. Algebra*</span> **47** (9), 3500–3519 (2019)

D. Gonçalves, D. Royer, Leavitt path algebras as partial skew group rings, <span>*Comm. Algebra*</span> <span>**42**</span> (8), 3578–3592 (2014)

D. Gonçalves and G. Yoneda, Free path groupoid grading on Leavitt path algebras, <span>*Internat. J. Algebra Comput.*</span> <span>**26**</span> (6), 1217–1235 (2016)

D. Lännstrom, Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras, *Algebr. Represent. Theory* <span>**23**</span>, 1707– 1726 (2020)

D. Lännstrom, A characterization of graded von Neumann regular rings with applications to Leavitt path algebras, *J. Algebra* <span>**567**</span> (2021) 91– 113.

P. Lundström, The category of groupoid graded modules, <span>*Colloq. Math.*</span> **100** (4), 195–211 (2004)

J. C. Mc Connel, J. C. Robson, <span>*Noncommutative Noetherian Rings,*</span> Pure Appl. Math., John Wiley and Sons, Chichester (1988)

P. Nystedt, J. Öinert and H. Pinedo, Artinian and noetherian partial skew groupoid rings, <span>*J. Algebra*</span> **503**, 433–452 (2018)

P. Nystedt, J.Öinert, H. Pinedo, Epsilon-strongly graded rings, separability and semisimplicity, <span>*J. Algebra*</span> **514**, 1–24 (2018)

P. Nystedt, J.Öinert, H. Pinedo, Epsilon-strongly groupoid-graded rings, the Picard inverse category and cohomology, <span>*Glasgow Math. J.*</span> **62**, 233–259 (2020)

J. K. Park, Artinian skew group rings, <span>*Proc. Am. Math. Soc*</span> **75** (1), 1–7 (1979)

D. S. Passman, <span>*The algebraic structure of group rings,*</span> Wiley, New York (1977)

H. Yahya, A note on graded regular rings, <span>*Comm. Algebra*</span>, **25** (1), 223-228 (1997)

[^1]: <span>**Mathematics Subject Classification 2010**</span>: Primary 16W22, 16S35, 16P20. Secondary 16W50, 16W55

[^2]: <span>**Keywords and phrases:**</span> Partial skew groupoid ring, group-type partial action, Leavitt path algebras

{'source': 'arXiv_out/2103'}