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 Advances in Pure Mathematics, 2013, 3, 666-669 Published Online November 2013 (http://www.scirp.org/journal/apm) http://dx.doi.org/10.4236/apm.2013.38089 Open Access APM The p.q.-Baer Property of Skew Group Rings under Finite Group Action* Bo Li, Hailan Jin# Department of Mathematics, College of Sciences, Yanbian University, Yanji, China Email: #[email protected], [email protected] Received October 14, 2013; revised November 14, 2013; accepted November 20, 2013 Copyright © 2013 Bo Li, Hailan Jin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper, Let R is a ring, G be a finite group of ring automorphisms of R. R*G denote the skew group ring of R un- der G. We investigate the right p.q.-Baer property of skew group rings under finite group action, Assume that R is a semiprime ring with a finite group G of X-outer ring automorphisms of R, then 1) R*G is p.q.-Baer if and only if R is G-p.q.-Baer; 2) if R is p.q.-Baer, then R*G is p.q.-Baer. Keywords: p. q.-Baer Proper t y ; Skew Gro u p Ring; Group Action 1. Introduction Throughout this paper all rings are associative with iden- tity unless otherwise stated. Let R is a ring, for a non- empty subset X of a ring R, (resp., R rX R lX) denote a right (resp.,left) annihilator of X in R. A ring R is called right principally quasi-Baer (simply, right p.q.- Baer) if the right annihilator of every principal right id eal of R is generated, as a right ideal by an idempotent of R in [1]. A left principally quasi-Baer (simply, left p.q.- Baer) ring is defined similarly. Right p.q.-Baer rings have been initially studied in [1]. For more details on (right) p.q.-Baer rings, see [1-6]. A ring R is called quasi-Baer if the right annihilator of every right ideal is generated, as a right ideal by an idempotent of R in [7] (see also [8]. A ring R is called biregular, if for each x R, for some central idempotent RxR eReR . We note that the class of right p.q.-Baer rings is a gener- alization of classes of quasi-Baer rings and biregular rings. denote a fixed maximal right ring of quo- tients of R. Recall from [9] an idempotent e of a ring R is called left (resp., right) semicentral if (resp., ) for all . Equivalently, an idempotent e is left (resp., right) semicentral if and only if (resp., ) is a two-sided ideal of R. (resp., QR ae eae eR ea eae Re aR l SR r SR) denote the set of all left (resp., right) semicentral idem- potents. An idempotent e of a ring R is called semicentral reduced if 0, l SeRe e. According to [2] a ring R is called semicentral reduced if , i.e., 1 is a semicentral reduced idempotent of R. 0,1 l SR If R is a semiprime ring and I is a two-sided ideal of R, then RR lIrI . For a right R-module M and a sub- module N of M, we use ess R R and NMden R R NM to denote that NR is essential in MR and NR is dense in MR, respectively. Let R is a ring, Aut R RG d enote a group of ring auto- morphisms of R, G be a subgroup of . Aut R gG Rg The skew group ring R*G is defined to be ,ab R with addition given component wise and multiplication given as follows: if and , g hG, then ab bha 0 1 ggh Rgh b. We begin with the following example. 2. Preliminary Example 2.1 There exist a ring R and a finite group G of ring automorphisms of R such that R is right p.q.-Baer but R*G is not right p.q.-Baer. Let F F R F with a field F of characteristic 2, then R is right p.q.-Baer. De fi ne Aut g R by 1 11 11 00100 ab ab gcc 1 *Project supported by the National Natural Science Foundation o f China (11361063). #Corresponding author. .  B. LI, H. L. JIN 667 Since characteristic of F is 2, Then . 21g Now we show that R*G is not right p.q.-Baer. Con- sider the right ideal 1 g RG of R*G generated by 1 g . By computation, we have 1 , 000 0 RG rgRG xy xxy g xy F G Suppose that 1 RG rgRGeR for some . Note that the idempotents of R*G are 0, 1. 2 ee RG 10 00 00 ab g , 00 01 00 ab g with . Since ,ab F 1 RG er gRG , the only possible choice for e is 0. Thus if R*G is right p.q.-Baer, then it follows that . This is a 10 RG rgRG contradiction. Therefore R*G is not right p.q.-Baer. Also we see that R*G is not left p.q.-Baer. Definition 2.2 Let R be a semiprime ring. For Aut g R, let for each g gm x QRxr rxrR , where is the Martindale right ring of quotients of R (see [10] for more on ). We say that m QR m QR g is X-outer if 0 g . A subgroup G of is called X-outer on R if every 1 Aut R g G is X-outer. Assume that R is a semiprime ring, then for Aut g R, let for each g g x QR xrrxrR . For Aut g R, we claim that g g . Obviously g g . Conversely, if g x then x RRx. There exists den R R I R such that x IR. Therefore , RI R den R R RI R, and x RI RxIR. Thus m x QR, hence g x . Theref or e g g . So if G is X-outer on R, then G can be considered as a group of ring automor- phismms of and G is X-outer on QR QR. For more details for X-outer ring automorphisms of a ring, etc., see [10, p. 396] and [11]. We say that a ring R has no nonzero-torsion ( is a positive integer) if with implies n Rn a0na a0 . Lemma 2.3 Let R be a semiprime ring and G a group of ring automorphisms of R. 1) [11,12] If G is X-outer, then every nonzero two- sided ideal of R*G intersects R nontrivially. Hence R*G is semiprime. 2) [11] If G is finite and R has no nonzero G-to rsion, Then R*G is semiprime. For a ring R, we use to denote the center of R. Cen R Lemma 2.4 For a semiprime ring R, let G be a group of X-outer ring au tomorphisms of R. Then CenCen G RG R . Proof. Let 122 1C nn aag agRen with i aR , 1 the identity of G, and i g G . The 122 122 11 nn nn aag agbbaag ag for all bR . So 11 ab ba , for all b 2 22 ,, n g gnn ab baab ba R . Since G is X-outer, it follows that 20an a . Hence 11 1aaR . Also since bb for all bR , we have that 1CenaR. Note that for all g G , implies 1 g 111 ag gaag 1 11 g aa . So . Thus 1Cena G R G CenCen G RG R . Conversely, is clear. Cen Cen G RR Therefore Cen CenRGR G. Lemma 2.5 [13,14] Let R be a ring and G a finite group of ring automorphisms of R. Then QR G is the maximal right ring of quatients of R. Assume that a group G of ring automorphisms of a ring R is finite. Then for , let aR g gG tr aa , which is called the trac e of a. Also for a right ideal I of R, the right ideal g gG tr IaaI of RG is called the trace of I. Say 1,, n Gg g. we put 1n tg g RG . For rR and 112 2nn ag agag RG with i aR , define 11 1nn g g gg n. Then R is a right R*G-module. Moreover, we see that rra ra G R G RR is an G, G RR -bimodule. Lemma 2.6 Assume that R is a semiprime ring and eBQR ess . Let I be a two-sided ideal of R such that R R I eR and R rI fR with f BR. Then 1ef . Proof. Since R is semiprime, 1 ess RRR R I lrI fR. Thus 1 ess R R I fQR . As ess R R I eR, ess R R I eQ R. We note that e and 1f are in BQR. So we have that . 1ef Proposition 2.7 [1] Let R be a semiprime ring. Then the following are equivalent. 1) R is right p.q.-Baer. 2) Every principal two-sided ideal of R is right essen- tial in a ring direct summand of R. Open Access APM  B. LI, H. L. JIN 668 3) Every finitely generated two-sided ideal of R is right essential in a ring direct summand of R. 4) Every principal two-sided ideal of R that is closed as a right ideal is a direct summand of R. 5) For every principal two-sided ideal I of R, R rI is right essential in a direct summand of R. 6) R is left p.q.-Baer. For a ring R with a group G of ring automorphisms of R, we say that a right ideal I of R is G-invariant if g I I for every g G, where gg I aa I. As- sume that R is a semiprime ring with a group G of ring automorphisms of R. We say that R is G-p.q.-Baer if the right annihilator of every finitely generated G-invariant two-sided ideal is generated by an idempotent, as a right ideal. By Proposition 2.7, if a ring R is semiprime p.q.- Baer with a group G of ring automorphisms of R, then R is G-p.q.-Baer. A ring R is called right Rickart if the right annihilator of each element is generated by an idempotent of R. A left Rickart ring is defined similarly. A ring R is called Rickart if R is both right and left Rickart. A ring R is said to be reduced if R has no nonzero nilpotent ele ment. We note that reduced Rickart rings are p.q.-Baer rings. We put there exists with p ess R R BQR eBQRxR RxReR Let be the subring of generated by ˆpqB QR BQR QR R and . p Lemma 2.8 [15] Assume that R is a semiprime ring. Then: 1) The ring is the smallest right ring of quotients of R whic h is p .q.-Bae r. ˆpqB QR 2) R is p.q.-Baer if and only if . p BQR R With these preparations, in spite of Example 2.1, we have the following result for p.q.-Baer property of R*G on a semiprime ring R for the case when G is finite and X-outer. 3. Main Results Theorem 3.1 Let R be a semiprime ring with a finite group G of X-outer ring automorphisms of R. Then R*G is p.q.-Baer if and only if R is G-p.q.-Baer. Proof. Assume that R*G is p.q.-Baer. Say 1n I Ra RRaR is a finitely generated G-invariant two-sided ideal of R with i. Then aR I G is a two-sided ideal of R*G. Moreover, 1, n I GRGaRGRGa RG Note that R*G is semiprime by Lemma2.3, So Propo- sition 2.7 yields that there exists such that l eSRG ess R GR IGeRG G . Since R*G is semipr ime, eBRG G by [9]. Hence by Lemma2.4, . First, we see that CeneR ess R R I eR. For this, let 0 with r er eRR . As ess R G eRGIG RG , there exists RG such that 0erI G . Say 11 nn bg bg with and i bRi g G for 1,i, .n Then 11 nn ererb gerbgI G . Hence 0j erb I for some , so jess R R I eR. As G 2Cenee R, I eRe, and so ess eRe eRe I eRe. Now we show that 1 R rI eR. If 0e , then R rI R . S o we may a ss ume t ha t . Note that eRe 0e is semiprime and ess eRe eRe I eRe, and so 0 eRe rI . Hence 0 RR eRr IeRe r I . As , IeR 1R eRr I . From the modular law, 1 RR rIeReRrI . But since 0 R eRr I , . There- fore R is G-p.q.-Baer. 1 R rI eR Conversely, let R be G-p.q.-Baer. Take p eBQRG . Then Cen G eQR by Lemma 2.4 since G is also X-outer on as was noted. Also there exists QR RG such that ess R GR RGRGeRG G because QRG is the maximal right ring of quotients of RG (Lemma 2.5) and . Say p eBQRG 1122nn ag agag with and i aRi g G for 1, 2,,in . Then eR GeRG R and so i ae for each 1, 2,,in . Consider 1, g i ig G n K RaR . Then K is a finitely generated G-invariant two-sided ideal of R. Further, K eR because . By Cen G eQR ess the preceding argument, we see that R R K eR. From the assumption, there exists l f SRBR such that R rK fR. Thus 1efR G by Lemma 2.6. There- fore eR R , so . From pRGRGBQ Open Access APM  B. LI, H. L. JIN Open Access APM 669 [7] W. E. Clark, “Twisted Matrix Units Semigroup Alge- bras,” Duke Mathematical Journal, Vol. 34, No. 3, 1967, pp. 417-423. http://dx.doi.org/10.1215/S0012-7094-67-03446-1 Lemma2.8, R*G is p.q.-Baer. Corollary 3.2 Let R be a semiprime ring with a finite group G of X-outer ring automorphisms of R. If R is p.q.-Baer, then R*G is p.q.-Baer. [8] A. Pollingher and A. Zaks, “On Baer and Quasi-Baer Rings,” Duke Mathematical Journal, Vol. 37, No. 1, 1970, pp. 127-138. http://dx.doi.org/10.1215/S0012-7094-70-03718-X Proof. The proof follows immediately by Theorem 3.1. 4. Conclusion [9] G. F. Birkenmeier, “Idempotents and Completely Semi- prime Ideals,” Communications in Algebra, Vol. 11, No. 6, 1983, pp. 567-580. http://dx.doi.org/10.1080/00927878308822865 In [16] researched quasi-Baer property of skew group rings under finite group actions on a semiprime ring and their applications to C*-algebras (see also [17,18]). In this paper, we investigate the right p.q.-Baer property of skew group rings under finite group action. Assume that is a semiprime ring with a finite group G of X-outer ring automorphisms of R, then 1) R*G is p.q.-Baer if and only if R is G-p.q. -Baer; 2) ifis p.q.-Baer, then R*G is p.q.-Baer. R R [10] T. Y. Lam, “Lectures on Modules and Rings,” Springer, Berlin, 1998. [11] J. W. Fisher and S. 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