数理解析研究所講究録 1222 巻 2001 年 5-10

5ON ORDERED MONOID RINGS

岡山大学理学部 平野康之 (Yasuyuki Hirano)

Faculty of Science, Okayama University,Amonoid $M$ is said to be ordered if the elements of $M$ are linearly oredered with respect to the relation
that, for all $x,y$ , $z\in Ci$ , $<y$ implies$<\mathrm{a}\mathrm{n}\mathrm{d}$$<zy$ and $xz$ $<yz$. It is$x$

known that torsion-free nilpotent groups andfree groups are ordered groups (see [8, Lemma 13.1.6 and Corollary 13.2.8]).
Hence any submonoid of atorsion-free nilpotent group or afree group is an$zx$$\mathrm{w}\mathrm{e}\mathrm{L}$ordered monoid.

Let$R$

be aring. Aleft (resp. right) annihilator of asubset $U$ of $R$ isdefined by $l_{R}(U)=\{a\in R | aU=0\}(resp. r_{R}(U)=\{a\in R |Ua=0\})$ .
Letbe an ordered monoid. Put $rAnn_{R}(2^{R})=\{r_{R}(U)|U\subseteq R\}$ and$G$$lAnn_{R}(2^{R})=\{l_{R}(U)|U\subseteq R\}$$r_{R}(U)RG$. Hence we have amap

by $\Phi(I)=I(RG)$ for every$I$. If$\Phi$$U$

is asubset of $R$ , then $r_{RG}(U)=$: $rAnn_{R}(2^{R})arrow rAnn_{RG}(2^{RG})$ defined$\in rAnn(R)$. For an element $f\in RG$ ,$C_{f}$denotes the set of coefficients of and for asubset of $RG$ ,denotes the. Then $r_{RG}(V)\cap R$ $=r_{R}(V)=r_{R}(Cv)$ . Hence we also have amap : $rAnn_{RG}(2^{RG})arrow rAnn_{R}(2^{R})$ defined by $\Psi(I)=I$for every$V$$f$$C_{V}$$\mathrm{s}\mathrm{e}\mathrm{t}\cup f\in vCf$$\Psi$$I$ $\in\Delta$. Obviously$\cap R$$\Phi$is injective and$\Psi$

is surjective. Clearly$\Phi$

is surjective if

6is injective, and in this case andother. We consider when is surjective.
Rege and Chhawchharia [10] aring
and only if$\Phi$$\Psi$$\Psi$

are the inverses of each$R$

is called an A rmendariz$\Phi$$\mathrm{F}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{i}\cdot \mathrm{g}$ring if whenever polynomials $f(x)=\Sigma_{=0}^{\pi\iota}\alpha.x^{:},g(x)=\Sigma_{j=0^{b}j^{X^{j}}}^{n}\in R[x]$ ,and . This name isconneted with the work of Armendariz [2]. Some results of Armendariz rings
can be found in [1], [5], [7] and [10]. Let be an ordered monoid. Aringsatisfy $f(x)g(x)=0$ we have $\alpha.b_{j}=0$ for every

$i$

$j$

$R$

$G$

is case a -Armendariz ring if whenever $p= \sum_{g\in G}a_{g}g,q=\sum_{h\in G}b_{h}h\in RG$
satisfy $pq=0$ we have $a_{g}b_{h}=0$ for every and in .
The folowing proposition shows that is bijective if and only if is$G$

$h$

$g$

$G$

$R$

$\Phi$

Armendariz.
Proposition 1. Let $R$ be aring and letstatements are equivalent:$G$

be an ordered monoid. The$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{i}\cdot \mathrm{g}$1) $R$ is G-Armendariz.

2) $rAnn_{R}(2^{R})$ $arrow rAnn_{RG}(2^{RG});Aarrow A(RG)$ is bijedive.
3) lAnnR(2R) $arrow lAnn_{RG}(2^{RG});Barrow(RG)B$ is bijective.

Folowing Kaplansky [6], aring is called aBaer ring if the left annihilator of each subset is generated by an idempotent. We note that the definitionof Baer rings is left-right symmetric. Aring $R$ is called aleft (resp. right)$R$. .-ring if the left (resp. right)$p$ $p$. ring is celled a . ring. Usingby an idempotent. Aleft and right
Proposition 1, we can generalize [7, Theorems 9and 10] as follows:$\mathrm{p}.\mathrm{p}$$\mathrm{p}.\mathrm{p}$$)$

be an ordered monoid and let be a. ring) if and only if $RG$ is aBaer ring
is aBaer ring (resp.

Corollary 2. Let

ring. Then
(resp.

$\mathrm{p}.\mathrm{p}$$R$. ring)

$R$

$G$

$\mathrm{p}.\mathrm{p}$$\mathrm{G}- \mathrm{A}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{d}\pi \mathrm{i}\mathrm{z}$7Aring$R$is called aG-quasi-Armendariz ring if whenever,satisfy $pRGq=0$ , then we have OiRbj $=0$ for every$p=\Sigma_{=0}^{n}.\cdot.\alpha.g.\cdot$$q=\Sigma_{j=0}^{n}b_{\mathrm{j}}h_{j}\in RG$

and. In case $G=\{x^{:}|i=0,1,2, \cdots\}$ , a-Armendariz ring is
simply called aquasi-Armendariz ring. In [5], we studied quasi-Armendariz
rings. Let $rAnn_{R}(id(R))$ (resp. $lAnn_{R}(id(R))$ ) denote the set { $r_{R}(U)|U$ is
an ideal of } (resp. $\{l_{R}(U)|U$ is an ideal of).
$i$

$j$

$R$

$G-\mathrm{q}\iota \mathrm{l}\mathrm{a}\mathrm{e}\mathrm{i}$$R\}$

Proposition 3. Let $R$ be aring and let $G$ be an ordered monoid. Then thefollowing statements are equivalent:
1) $R$ is G-quasi-Armendariz.
2) $rAnn_{R}(id(R))arrow rAnn_{RG}(id(RG));Aarrow A(RG)$ is bijective.
3) $lAnn_{R}(id(R))arrow lAnn_{RG}(id(RG));Barrow(RG)B$ is bijective.

Asubmodule $N$ of aleft$L\otimes_{R}Narrow L\otimes_{R}M$-module $M$ is cffied a pure submodule if$R$

is amonomorphism for every right $R$-module$L$.

be an ordered monoid. Then the following are equivalent;(1) $l_{R}(Ra)$ is pure as aleft ideal in $R$ for any element $a\in R$ ;Theorem 4. Let$G$(1) $l_{RG}(RGz)$ is pure as aleft ideal in $RG$ for any element $z\in RG$ ;In this case, $R$ is aG-quasi-Armendariz ring.
Corolary 5. Let $R$ be acommutative ring and let

monoid. Then each principal ideal of$R$$G$

be an abelian orderedis flat if and only if each principal

ideal of $RG$ is flat. In this case $R$ is a -Armendariz ring.
$G$

Aringof$R$$R$

is called quasi-Baer if the leftof every left ideal

is generated by an idempotent. Note that this definition is left-rightsymmetric. Some results of aquasi-Baer ring can be found in [3] and [9]8
Let R be aquasi-Baer ring and let a $\in R$ . Then $l_{R}(aR)=Re$ for some
idempotent $e\in R$, and so $l_{R}(aR)$ is pure as aleft ideal in . Therefore aquasi-Baer ring satisfies the hypothesis of Theorem 4. Hence we obtain the$R$

following.
be an ordered monoid. Aring
quasi-Baer ring if and only if $RG$ is aquasi-Baer ring. In this case,
Corolhry 6([4, Theorem 1]). Let

$G$

$R$

is a

$R$

is a

quasi-Armendariz ring.

Now we consider some extensions of G-quaei-Amendniz rings. Let $R$ bearing and let be apositive integer. Let . (R) denote the ring ofdenotes the ring
matrices over $R$ and .denote the $(i,j)$-matrix unit.
$n$$n\mathrm{x}$$\mathrm{M}_{\mathrm{s}}$$n$$T_{\mathfrak{n}}(R)$$\mathrm{q}_{j}$of all$n$$\mathrm{x}n$upper triangular matrices overTheorem 7. Letand letThen$G$.

be an ordered monoid. If$R$be subring of $M_{n}(R)$ such thatis also aG-quasi-Armendariz ring.

$S$

$S$

$R$

$q..\cdot Se_{jj}$aG-quaei-Amendariz ring$\subseteq S$

for aU:, $j\in\{1, \cdots,n\}$ .

be an ordered monoid. If aG-quasi-Armendariz ring,is also aG-quasi-Armendariz ring.
then, for any positive integer ,Corollary 8. Let

$R$

$G$

$T_{1}.(R)$$n$of is the ideal of generated by the. Indenotes the idealof $RG$ ,case $G=\{x^{:}|i=0,1,2, \cdots\}$ , acommutative ring is Gaussian if $A_{fg}=$
For $f\in RG7$ the content
coefficients of . For any subset$A_{f}$

$f$

$S$

$R$

$f$

$A_{S}$$\sum_{f\in S}A_{f}$$R$

for all $f,g\in R[x]$ (See [1]). We extend this notion to noncommutativerings as follows. Aring $R$ is said to be G-quasi-Gaussian if $A_{fRg}=A_{f}A_{g}$ for$A_{f}A_{g}$ffi $f,g\in RG$ .

Theorem 9. Aring$R$is G-quasi-Gaussian if and only if every homomorphi

9
image of $R$ is quasi-Armendariz.
Example 10. Aring $R$ is ftdly idempotent if $p=I$ for every twosided

ideal I of $R$ . Obviously aring $R$ is fuly idempotent if and only if every
homomorphic image of $R$ is semiprime. Von Neumann regular rings are fullyidempotent. We can easily see that asemiprime ring is G-quasi-Armendariz.

Therefore by Theorem 9, afully idempotent ring is aG-quasi-Gaussian ring.

References[1] D. D. ANDERSON AND V. CAMILLO, Armendariz rings and Gaussianrings Comm. in Algebra 26 (7)(1998), 2265-2272.
[2] E.

P. ARMENDARIZ, Anote on extensions of Baer and p.p. rings,

Australian Math. Soc. 18 (1974), 470-473.
[3] W. E. CLARK, Twisted matrix units semigroup algebras, Duke Math.

J. 34 (1967), 417-424.
[4] Y. HiRANO, On totaly ordered monoid rings over aquasi-Baer ring,to appear in Comm. Algebra.
[5] Y.

HiRANO, On annihilator ideals of anoncommutative polynomialring, to appear in J. Pure and Appl. Algebra.
[6] I. KAPLANSKY, Rings of Operators, Math. Lecture Notes Series, Benjarnin, , New York, 1965

10[7] N. K. KIM AND Y. LEE, Annendariz rings and reduced rings, J.

Algebra 223(20M),$477\triangleleft 88$.

[8] D. S. PASSMAN, The Algebraic Structure of Group Rings, John Wiley&Sons Ltd., 1977.
[9] P. POLLINGHER AND A. Zaks, On Baer and quasi-Baer rings, Duke

Math. J. 37(1970), 127-138.
[10] M. B. REGE AND S. CHHAWCHHARIA, Annendariz rings, Proc. Japan

Acad. Ser. A Maffi. Sci. 73(1997), 14-17