# TheL-Total Graph of an L-Module

Authors: Reza Ebrahimi Atani
Journal: ISRN Discrete Mathematics
 (2011)

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## Abstract

LetL be a complete lattice. We introduce and investigate
the L-total graph of an L-module over an L-commutative ring. The main
purpose of this paper is to extend the definition and results given in (Anderson and Badawi, 2008) to
more generalize the L-total graph of an L-module case.


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## Body

## 1. Introduction

It was Beck (see [1]) who first introduced the notion of a zero-divisor graph for commutative rings. This notion was later redefined by Anderson and Livingston in [2]. Since then, there has been a lot of interest in this subject, and various papers were published establishing different properties of these graphs as well as relations between graphs of various extensions (see [2–5]). Let R be a commutative ring with Z(R)being its set of zero-divisors elements. The total graph of R, denoted by T(Γ(R)), is the (undirected) graph with all elements of R as vertices, and, for distinct x,y∈R, the vertices x and y are adjacent if and only if x+y∈Z(R). The total graph of a commutative ring has been introduced and studied by Anderson and Badawi in [3]. In [6], the notion of the total torsion element graph of a module over a commutative ring is introduced.In [7], Zadeh introduced the concept of fuzzy set, which is a very useful tool to describe the situation in which the data is imprecise or vague. Many researchers used this concept to generalize some notions of algebra. Goguen in [8] generalized the notion of fuzzy subset of X to that of an L-subset, namely, a function from X to a lattice L. In [9], Rosenfeld considered the fuzzification of algebraic structures. Liu [10] introduced and examined the notion of a fuzzy ideal of a ring. Since then several authors have obtained interesting results on L-ideals of a ring R and L-modules (see [11, 12]). Also, L-zero-divisor graph of an L-commutative ring has been introduced and studied in [13].In the present paper we introduce a new class of graphs, called theL-total torsion element graph of a L-module (see Definition 2.2), and we completely characterize the structure of this graph. The total torsion element graph of a module over a commutative ring and the L-total torsion element graph of a L-module over a L-commutative ring are different concepts. Some of our results are analogous to the results given in [6]. The corresponding results are obtained by modification, and here we give a complete description of the L-total torsion element graph of an L-module.For the sake of completeness, we state some definitions and notation used throughout. For a graphΓ, by E(Γ) and V(Γ), we denote the set of all edges and vertices, respectively. We recall that a graph is connected if there exists a path connecting any two distinct vertices. The distance between two distinct vertices a and b, denoted by d(a,b), is the length of the shortest path connecting them (if such a path does not exist, then d(a,a)=0 and d(a,b)=∞). The diameter of a graph Γ, denoted by diam(Γ), is equal to sup{d(a,b):a,b∈V(Γ)}. A graph is complete if it is connected with diameter less than or equal to one. The girth of a graph Γ, denoted gr(Γ), is the length of the shortest cycle in Γ, provided Γ contains a cycle; otherwise, gr(Γ)=∞. We denote the complete graph on n vertices by Kn and the complete bipartite graph on m and n vertices by Km,n (we allow m and n to be infinite cardinals). We will sometimes call a K1,m a star graph. We say that two (induced) subgraphs Γ1 and Γ2 of Γ are disjoint if Γ1 and Γ2 have no common vertices and no vertex of Γ1 (resp., Γ2) is adjacent (in Γ) to any vertex not in Γ1 (resp., Γ2).LetR be a commutative ring, and L stands for a complete lattice with least element 0 and greatest element 1. By an L-subset μ of a nonempty set X, we mean a function μ from X to L. If L=[0,1], then μ is called a fuzzy subset of X. LX denotes the set of all L-subsets of X. We recall some definitions and lemmas from the book [12], which we need for development of our paper.Definition 1.1.
AnL-ring is a function μ:R→L, where (R,+,.) is a ring, which satisfies the following.(1)
μ≠0.(2)
μ(x-y)≥μ(x)∧μ(y) for every x,y in R.(3)
μ(xy)≥μ(x)∨μ(y) for every x,y in R.Definition 1.2.
Letμ∈LR.Then μ is called an L-ideal of R if for every x,y∈R the following conditions are satisfied. (1)
μ(x-y)≥μ(x)∧μ(y).(2)
μ(xy)≥μ(x)∨μ(y).
The set of allL-ideals of R is denoted by LI(R).Definition 1.3.
Assume thatM is an R-module, and let μ∈LM. Then μ is called an L-fuzzy R-module of M if for all x,y∈M and for all r∈R the following conditions are satisfied.(1)
μ(x-y)≥μ(x)∧μ(y).(2)
μ(rx)≥μ(x).(3)
μ(0M)=μ(1).
The set of allL-fuzzy R-modules of M is denoted by L(M).Lemma 1.4.
LetM be a module over a ring R, and μ∈L(M). Then μ(m)≤μ(0M) for every m∈M.


## 2.T(μ) Is a Submodule of M

LetM be a module over a commutative ring R, and let μ∈L(M). The structure of the L-total torsion element graph T(Γ(μ)) may be completely described in those cases when μ-torsion elements form a submodule of M. We begin with the key definition of this paper.Definition 2.1.
LetM be a module over a commutative ring R, and let μ∈L(M). A μ-torsion element is an element m∈M with μ(m)≠μ(0M) for which there exists a nonzero element r of R such that μ(rm)=μ(0M).The set ofμ-torsion elements in M will be denoted by T(μ).Definition 2.2.
LetM be a module over a ring R, and let μ∈L(M). We define the L-total torsion element graph of an L-module T(Γ(μ)) as follows: V(T(Γ(μ)))=M, E(T(Γ(μ)))={{x,y}:x+y∈T(μ)}.Notation 1.
For theμ-torsion element graph T(Γ(μ)), we denote the diameter, the girth, and the distance between two distinct vertices a and b, by diam(T(Γ(μ))), gr(T(Γ(μ))), and dμ(a,b), respectively.Remark 2.3.
LetMbe a module over a ringR, and letμ∈L(M). Clearly, ifμis a nonzero constant, thenT(Γ(μ))=∅. So throughout this paper, we will assume, unless otherwise stated, thatμis not a nonzero constant. Thus, there is a nonzero elementyofMsuch thatμ(y)≠μ(0M).We will useTof(μ) to denote the set of elements of M that are not μ-torsion elements. Let Tof(Γ(μ)) be the (induced) subgraph of T(Γ(μ)) with vertices Tof(μ), and let Tor(Γ(μ)) be the (induced) subgraph of T(Γ(μ)) with vertices T(μ).Definition 2.4.
LetM be a module over a ring R, and μ∈L(M). One defines the set annμ(M) by annμ(M)={r∈R:μ(rM)={μ(0M)}}, the μ-annihilator of M.Lemma 2.5.
LetM be a module over a ring R, and let μ∈L(M). Then annμ(M) is an L-ideal of R.Proof.
Letr,s∈annμ(M) and t∈R. If m∈M, then we have μ((r-s)m)≥μ(rm)∧μ(-sm)=μ(0M)∧μ(0M)=μ(0M) and μ(trm)=μ(t(rm))≥μ(rm)=μ(0M). It then follows from Lemma 1.4 that μ((r-s)m)=μ(0M); hence r-s∈annμ(M). Similarly, rt∈annμ(M).Theorem 2.6.
LetM be a module over a ring R and let μ∈L(M). Then the L-torsion element graph T(Γ(μ)) is complete if and only if T(μ)=M.Proof.
IfT(μ)=M, then for any vertices m,m′∈M, one has m+m′∈T(μ); hence they are adjacent in T(Γ(μ)). On the other hand, if T(Γ(μ)) is complete, then every vertex is adjacent to 0. Thus,m=m+0∈T(μ) for every m∈M. This completes the proof.Theorem 2.7.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is a submodule of M. Then one has the following.(i)
Tor(Γ(μ)) is a complete (induced) subgraph of T(Γ(μ)) and Tor(Γ(μ)) is disjoint from Tof(Γ(μ)).(ii)
If annμ(M)≠0, then T(Γ(μ)) is a complete graph.Proof.
(i)Tor(Γ(μ)) is complete directly from the definition. Finally, if m∈T(μ) and m′∈Tof(μ) were adjacent, then m+m′∈T(μ); so this, since T(μ) is a submodule, would lead to the contradiction m′∈T(μ).
(ii) Letm∈M. we may assume that μ(m)≠μ(0M). By assumption, there exists 0≠s∈R with μ(sM)=μ(0M), so μ(sm)=μ(0M). Thus m∈T(μ), and; therefore, T(Γ(μ)) is a complete graph by Theorem 2.6.Theorem 2.8.
LetM be a module over a ring R, and let μ∈L(M). Then T(Γ(μ)) is totally disconnected if and only if R has characteristic 2 and T(μ)={0M}.Proof.
IfT(μ)={0M}, then the vertices m1 and m2 are adjacent if and only if m1=-m2. Then T(Γ(μ)) is a disconnected graph, and its only edges are those that connect vertices mi and -mi (we do not need a priori assumption that R has characteristic 2). Conversely, assume that T(Γ(μ)) is totally disconnected. Then 0+m∉T(μ) for every nonzero element m of M. Thus, T(μ)={0M}. Further, since m+(-m)=0, we have m=-m (so μ(2m)=μ(0M)) for every m∈M with μ(m)≠μ(0M) by the total disconnectedness of the graph T(Γ(μ)). As T(μ)={0M}, it follows that 2=1R+1R=0. Thus, char(R)=2.Proposition 2.9.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is a submodule of M. If m∈Tof(μ), then 2m∈T(μ) if and only if 2∈Z(R).Proof.
First suppose that2m∈T(μ). Since m∉T(μ), we get that μ(m)≠μ(0M), and, for all r∈R,μ(rm)=μ(0M) implies that r=0. Since 2m∈T(μ), there is a nonzero element c∈R such that μ(c(2m))=μ((2c)m)=μ(0M), and, since m∉T(μ), one must have 2c=0; hence, 2∈Z(R). Conversely, assume that 2∈Z(R). Then there exists 0≠d∈R with 2d=0. Since μ(0M)=μ((2d)m)=μ(d(2m)), we have 2m∈T(μ).Theorem 2.10.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is a proper submodule of M. Then T(Γ(μ)) is disconnected.Proof.
IfT(μ)={0M}, then T(Γ(μ)) is disconnected by Theorem 2.8. If T(μ)≠{0M}, then the subgraphs of Tor(Γ(μ)) and Tof(Γ(μ)) are disjoint by Theorem 2.7 (i), as required.Theorem 2.11.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is a proper submodule of M. Suppose |T(μ)|=α and |M/T(μ)|=β. Then one has the following.(i)
If2∈Z(R), then T(Γ(μ)) is a union of β disjoint complete graphs Kα.(ii)
If2∉Z(R), then T(Γ(μ)) is a union of (β-1)/2 disjoint bipartite graphs Kα,α and one complete graph Kα.Proof.
(i) Assume that2∈Z(R) and let m,m′∈Tof(μ) be such that m+T(μ)≠m′+T(μ). The elements m+t, m+t′ from the same coset m+T(μ) are adjacent if and only if 2m∈T(μ), so 2∈Z(R), according to the Proposition 2.9. Then m+t and m′+t′ are not adjacent (otherwise, we would have m-m′=m+m′-2m′∈T(μ)), and; therefore, m+T(μ)=m′+T(μ). Since every coset has cardinality α, we conclude that T(Γ(μ)) is the disjoint union of β complete graph Kα.
(ii) If2∉Z(R), then the elements m+t, m+t′ from m+T(μ) are obviously not adjacent. The elements m+t, m′+t′ from different cosets are adjacent if and only if m+m′∈T(μ) or m+T(μ)=(-m)+T(μ). In this way we obtain that the subgraph spanned by the vertices from Tof(μ) is a disjoint union of (β-1)/2 (=β if β is infinite) disjoint bipartite graph Kα,α.Proposition 2.12.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is a proper submodule of M. Then one has the following.(i)
Tof(Γ(μ)) is complete if and only if either |M/T(μ)|=2 or |M/T(μ)|=|M|=3.(ii)
Tof(Γ(μ)) is connected if and only if either |M/T(μ)|=2 or |M/T(μ)|=3.(iii)
Tof(Γ(μ)) and, hence; (Tor(Γ(μ)) and T(Γ(μ))) is totally disconnected if and only if T(μ)={0M} and 2∈Z(R).Proof.
Let|M/T(μ)|=β and |T(μ)|=α.(i)
LetTof(Γ(μ)) be complete. Then, by Theorem 2.11, Tof(Γ(μ)) is complete if and only if Tof(Γ(μ)) is a single Kα or K1,1. If 2∈Z(R), then β-1=1. Thus, β=2, and hence |M/T(μ)|=2. If 2∉Z(R), then α=1 and (β-1)/2=1. Thus, T(μ)={0} and β=3; hence, |M|=|M/T(μ)|=3. The reverse implication may be proved in a similar way as in [6, Theorem  2.6 (1)].(ii)
By theorem 2.11,Tof(Γ(μ)) is connected if and only if Tof(Γ(μ)) is a single Kα or Kα,α. Thus, either β-1=1 if 2∈Z(R) or (β-1)/2=1 if 2∉Z(R); hence, β=2 or β=3, respectively, as needed. The reverse implication may be proved in a similar way as in [3, Theorem  2.6 (2)].(iii)
Tof(Γ(μ)) is totally disconnected if and only if it is a disjoint union of K1’s. So by Theorem 2.11, |T(μ)|=1 and |M/T(μ)|=1, and the proof is complete.By the proof of the Proposition2.12, the next theorem gives a more explicit description of the diameter of Tof(Γ(μ)).Theorem 2.13.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is a proper submodule of M. Then one has the following.(i)
diam(Tof(Γ(μ)))=0 if and only if T(μ)={0} and |M|=2.(ii)
diam(Tof(Γ(μ)))=1 if and only if either T(μ)≠{0M} and |M/T(μ)|=2 or T(μ)={0} and |M|=3.(iii)
diam(Tof(Γ(μ)))=2 if and only if T(μ)≠{0M} and |M/T(μ)|=3.(iv)
Otherwise,diam(Tof(Γ(μ)))=∞.Proposition 2.14.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is a proper submodule of M. Then gr(Tof(Γ(μ)))=3,4 or ∞. In particular, gr(Tof(Γ(μ)))≤4 if Tof(Γ(μ)) contains a cycle.Proof.
LetTof(Γ(μ)) contain a cycle. Then since Tof(Γ(μ)) is disjoint union of either complete or complete bipartite graphs by Theorem 2.11, it must contain either a 3 cycles or a 4 cycles. Thus gr(Tof(Γ(μ)))≤4.Theorem 2.15.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is a proper submodule of M. Then one has the following.(i)
(a)
gr(Tof(Γ(μ)))=3 if and only if 2∈Z(R) and |T(μ)|≥3.(b)
gr(Tof(Γ(μ)))=4 if and only if 2∉Z(R) and |T(μ)|≥2.(c)
Otherwise,gr(Tof(Γ(μ)))=∞.(ii)
(a)
gr(T(Γ(μ)))=3 if and only if |T(μ)|≥3.(b)
gr(T(Γ(μ)))=4 if and only if 2∉Z(R) and |T(μ)|=2.(c)
Otherwise,gr(T(Γ(μ)))=∞.Proof.
Apply Theorem2.11, Proposition 2.14, and Theorem 2.7 (i).The previous theorems give a complete description of the structure of theL-total torsion element graph of an L-module M when T(μ) is a submodule. The question under what conditions T(μ) is a submodule of M and how is this related to the condition that Z(R) is an ideal in R naturally arises. We prove that the following results holds.Theorem 2.16.
LetM be a module over a ring R, and let μ∈L(M). Then one has the following.(i)
IfZ(R)={0R}, then T(μ) is a submodule of M.(ii)
IfZ(R)=Rc is a principal ideal of R with c a nilpotent element of R, then T(μ) is a submodule of M.Proof.
(i) Letm,m′∈T(μ) and r∈R. There are nonzero elements a,b∈R such that μ(m)≠μ(0M), μ(m′)≠μ(0M), and μ(am)=μ(bm′)=μ(0M) with ab≠0 (since R is an integral domain). It follows that μ(ab(m+m′))≥μ(abm)∧μ(abm′)=μ(0M)∧μ(0M)=μ(0M); hence, μ(ab(m+m′))=μ(0M) by Lemma 1.4. Thus, m+m′∈T(μ). Similarly, rm∈T(μ), and this completes the proof.
(ii) Assume thatT(μ) is not a submodule of M. Then there are elements m,m′∈T(μ) such that m+m′∉T(μ). By assumption, there exist nonzero elements r,s∈R such that μ(rm)=μ(0M)=μ(sm′)=μ(0M), where μ(m)≠μ(0M) and μ(m′)≠μ(0M). Then μ(rs(m+m′))=μ(0M) and m+m′∉T(μ), so we must have rs=0, and; thus, r,s∈Z(R). Since c is nilpotent, we have r=r1ct and s=s1cu, for some r1,s1∉Z(R). We may assume that t≥u. Then for the nonzero element s1r of R we have μ(s1r(m+m′))=μ(0M) which is contrary to the assumption that m+m′∉T(μ).Example 2.17.
Assume thatR=ℤ is the ring integers, and let M=R. We define the mapping μ:M→[0,1] by
(2.1)μ(m)={12ifx∈2Z,15otherwise.
Then μ∈L(M) and T(μ)=M. Thus, T(Γ(μ)) is a complete graph by Theorem 2.6.Example 2.18.
LetM1=R1=Z8 denote the ring of integers modulo 8 and M2=R2=Z25 the ring of integers modulo 25. We define the mappings μ1:M1→[0,1] by
(2.2)μ1(x)={1ifx=0̅,12otherwise
and μ2:M2→[0,1] by
(2.3)μ2(m)={1ifx=0̅,13otherwise.
Then, for each i (1≤i≤2), μi∈L(Mi), T(μ1)={0̅,2̅,4̅,6̅}, and T(μ2)={0̅,5̅,1̅0,1̅5,2̅0}. An inspection will show that T(μ1) and T(μ2) are submodules of M1 and M2, respectively. Therefore, by Theorem 2.11, we have the following results.(1)
Since2∈Z(R1), we conclude that T(Γ(μ1)) is a union of 2 disjoint K4.(2)
Since2∉Z(R2), we conclude that T(Γ(μ2)) is a disjoint union of 2 complete graph K5 and 5 bipartite K5,5.


## 3.T(μ) Is Not a Submodule of M

We continue to use the notation already established, soM is a module over a commutative ring R and μ∈L(M). In this section, we study the L-torsion element graph T(Γ(μ)) when T(μ) is not a submodule of M.Lemma 3.1.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is not a submodule of M. Then there are distinct m,m′∈T(μ)* such that m+m′∈Tof(μ).Proof.
It suffices to show thatT(μ) is always closed under scalar multiplication of its elements by elements of R. Let m∈T(μ) and r∈R. There is a nonzero element s∈R with μ(sm)=μ(0M) such that μ(m)≠μ(0M), so μ(s(rm))=μ(r(sm))≥μ(sm)=μ(0M); hence, μ(s(rm))=μ(0M) by Lemma 1.4, as required.Theorem 3.2.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is not a submodule of M. Then one has the following.(i)
Tor(Γ(μ)) is connected with diam(Tor(Γ(μ)))=2.(ii)
Some vertex ofTor(Γ(μ)) is adjacent to a vertex of Tof(Γ(μ)). In particular, the subgraphs Tor(Γ(μ)) and Tof(Γ(μ)) of T(Γ(μ)) are not disjoint.(iii)
IfTof(Γ(μ)) is connected, then T(Γ(μ)) is connected.Proof.
(i) Letx∈T(μ)*. Then x is adjacent to 0. Thus, x-0-y is a path in Tor(Γ(μ)) of length two between any two distinct x,y∈T(μ)*. Moreover, there exist nonadjacent x,y∈T(μ)* by Lemma 3.1; thus, diam(Tor(Γ(μ)))=2.
(ii) By Lemma3.1, there exist distinct x,y∈T(μ)* such that x+y∈Tof(μ). Then -x∈T(μ) and x+y∈Tof(μ) are adjacent vertices in T(Γ(μ)) since -x+(x+y)=y∈T(μ). Finally, the “in particular” statement follows from Lemma 3.1.
(iii) By part (i) above, it suffices to show that there is a path fromx to y in T(Γ(μ)) for any x∈T(μ) and y∈Tof(μ). By part (ii) above, there exist adjacent vertices c and d in Tor(Γ(μ)) and Tof(Γ(μ)), respectively. Since Tor(Γ(μ)) is connected, there is a path from x to c in Tor(Γ(μ)), and, since Tof(Γ(μ)) is connected, there is a path from d to y in Tof(Γ(μ)). Then there is a path from x to y in T(Γ(μ)) since c and d are adjacent in T(Γ(μ)). Thus, T(Γ(μ)) is connected.Proposition 3.3.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is not a submodule of M. If the identity of the ring R is a sum of n zero divisors, then every element of the M is the sum of at most nμ-torsion elements.Proof.
Letx∈M and r∈Z(R). We may assume that μ(x)≠μ(0M). Then there is a nonzero element b∈R such that rb=0, so μ(b(rx))=μ((rb)x)=μ(0M) with μ(rx)≠μ(0M). Therefore, if x∈M and r∈R, then rx∈T(μ), so, for all x∈M, 1=c1+⋯+cn implies that x=c1x+⋯+cnx, as needed.Theorem 3.4.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is not a submodule of M. Then T(Γ(μ)) is connected if and only if M is generated by its μ-torsion elements.Proof.
Let us first prove that the connectedness of the graphT(Γ(μ)) implies that the module M is generated by its μ-torsion elements. Suppose that this is not true. Then there exists x∈M which does not have a representation of the form x=x1+⋯+xn, where xi∈T(μ). Moreover, x≠0 since 0∈T(μ). We show that there does not exist a path from 0 to x in T(Γ(μ)). If 0-y1-y2-⋯-ym-x is a path in T(Γ(μ)), y1,y1+y2,…,ym-1+ym,ym+x are μ-torsion elements and x may be represented as x=(ym+x)-(ym-1+ym)+⋯+(-1)m-1(y1+y2)+(-1)my1. This contradicts the assumption that x is not a sum of μ-torsion elements. The reverse implication may be proved in a similar way as in [6, Theorem  3.2].We give here with an interesting result linking theL-torsion element graph T(Γ(μ)) to the total graph of a commutative ring T(Γ(R)).Theorem 3.5.
LetM be a module over a ring R, and let μ∈L(M). If T(Γ(R)) is connected, then T(Γ(μ)) is a connected graph. In particular, dμ(0,x)≤d(0,1) for every x∈M.Proof.
Note that, ifx∈M and r∈Z(R), then rm∈T(μ) (see Proposition 3.3). Now suppose that T(Γ(R)) is connected, and let x∈M. Let 0-s1-s2-⋯-sn-1 be a path from 0 to 1 in T(Γ(R)). Then s1,s1+s2,…,sn+1∈Z(R); hence, 0M-s1x-⋯-snx-x is a path from 0M to x. As all vertices may be connected via 0M, T(Γ(μ)) is connected.Theorem 3.6.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is not a submodule of M. If every element of M is a sum of at most nμ-torsion elements, then diam(T(Γ(μ)))≤n. If n is the smallest such number, then diam(T(Γ(μ)))=n.Proof.
We first show that, by assumption,dμ(0,x)≤n for every nonzero element x of M. Assume that x=x1+⋯+xn, where xi∈T(μ). Set yi=(-1)n+i(x1+⋯+xn) for i=1,…,n. Then 0-y1-y2-⋯-yn=x is a path from 0 to x of length n in T(Γ(μ)). Let u and w be distinct elements in M. We show that dμ(u,w)≤n. If (u-w)-z1-z2-⋯-zn-1 is a path from 0 to u-w and u+w-s1-s2-⋯-sn-1 is a path from 0 to u+w, then, from the previous discussion, the lengths of both paths are at most n. Depending on the fact whether n is even or odd, we obtain the paths
(3.1)u-(z1-w)-(z2+w)-⋯-(zn-1-w)-w
or u-(s1+w)-(s2-w)-⋯-(sn-1-w)-w from u to w of length n. Assume that n is the smallest such number, and let a=a1+a2+⋯+an be the shortest representation of the elements x as a sum of μ-torsion elements. From the previous discussion, we have dμ(0,x)≤n. Suppose that dμ(0,x)=k≤n, and let 0-t1-t2-⋯-tk-1-x be a path in T(Γ(μ)). It means, a presentation of the element x as a sum of k<nμ-torsion elements (see the proof of Theorem 3.4), which is a contradiction. This completes the proof.Corollary 3.7.
LetM be a module over a ring R, and let μ∈L(M) such that Z(R) is not an ideal of R and <Z(R)>=R. If diamT((Γ(R)))=n, then diamT((Γ(μ)))≤n. In particular, if R is finite, then diamT((Γ(μ)))≤2.Proof.
This follows from Proposition3.3 and Theorem 3.6. Finally, if R is a finite ring such that Z(R) is not an ideal of R, then diamT((Γ(R)))=2 by [3, Theorem  3.4], as required.By Lemma3.1, the following theorem may be proved in a similar way as in [6, Theorem  3.5].Theorem 3.8.
LetM be a module over a ring R, and let μ∈L(M) such that T(μ) is not a submodule of M. Then one has the following.(i)
Eithergr(Tor(Γ(μ)))=3 or gr(Tor(Γ(μ)))=∞.(ii)
gr(T(Γ(μ)))=3 if and only if gr(Tor(Γ(μ)))=3.(iii)
Ifgr(T(Γ(μ)))=4, then gr(Tor(Γ(μ)))=∞.(iv)
IfChar(R)≠2, then gr(Tof(Γ(μ)))=3,4 or ∞.Example 3.9.
LetM=R=Z6 denote the ring of integers modulo 6. We define the mapping μ:M→[0,1] by
(3.2)μ(x)={1ifx=0̅,14otherwise.
Then μ∈L(M) and T(μ)={0̅,2̅,3̅,4̅}. Now one can easily show that T(μ) is not a submodule of M and Tof(μ)={1̅,5̅}. Clearly, Tor(Γ(μ)) is connected with diam(Tor(Γ(μ)))=2. Moreover, since 1̅+3̅∈T(μ), we conclude that the subgraphs Tof(Γ(μ)) and Tor(Γ(μ)) of T(Γ(μ)) are not disjoint. Furthermore, T(Γ(μ)) is connected since Tof(Γ(μ)) is connected.



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*Source: 491936-2011-12-07.xml*