 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: #hljin98@ybu.edu.cn, hljin98@hanmail.net 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., RrXRlX) denote a right (resp.,left) annihilator ofX 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 . A left principally quasi-Baer (simply, left p.q.- Baer) ring is defined similarly. Right p.q.-Baer rings have been initially studied in . 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  (see also . A ring R is called biregular, if for each xR, for some central idempotent RxR eReR. 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  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., QRae eaeeRea eaeReaRlSRrSR) denote the set of all left (resp., right) semicentral idem-potents. An idempotent e of a ring R is called semicentral reduced if 0,lSeRe e. According to  a ring R is called semicentral reduced if , i.e., 1 is a semicentral reduced idempotent of R.  0,1lSRIf R is a semiprime ring and I is a two-sided ideal of R, then RRlIrI . For a right R-module M and a sub-module N of M, we use essRR and NMdenRRNM to denote that NR is essential in MR and NR is dense in MR, respectively. Let R is a ring, Aut RRG d enote a group of ring auto-morphisms of R, G be a subgroup of . Aut RgGRgThe skew group ring R*G is defined to be ,ab R with addition given component wise and multiplication given as follows: if  and ,ghG, then ab bha01ggh 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 FFRF with a field F of characteristic 2, then R is right p.q.-Baer. De fi ne AutgR by 111 1100100ab abgcc1   *Project supported by the National Natural Science Foundation of China (11361063). #Corresponding author.   . B. LI, H. L. JIN 667Since 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gRG of R*G generated by 1g. By computation, we have  1,000 0RGrgRGxy xxygxy F   G Suppose that  1RGrgRGeR for some . Note that the idempotents of R*G are 0, 1. 2ee RG1000 00abg, 0001 00abg    with . Since ,ab F1RGer 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  10RGrgRG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 AutgR, let  for each ggmxQRxr rxrR , where is the Martindale right ring of quotients of R (see  for more on ). We say that mQRmQRg is X-outer if 0g. A subgroup G of is called X-outer on R if every 1Aut RgG is X-outer. Assume that R is a semiprime ring, then for AutgR, let  for each ggxQR xrrxrR . For AutgR, we claim that gg . Obviously gg . Conversely, if gx then xRRx. There exists denRRIR such that xIR. Therefore , RI RdenRRRI R, and xRI RxIR. Thus mxQR, hence gx. Theref or e gg . 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 . We say that a ring R has no nonzero-torsion ( is a positive integer) if with implies nRna0na a0. 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)  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 RLemma 2.4 For a semiprime ring R, let G be a group of X-outer ring au tomorphisms of R. Then CenCen GRG R . Proof. Let 1221Cnnaag agRen  with iaR, 1 the identity of G, and igG. The 122 12211nn nnaag agbbaag ag  for all bR. So 11ab ba, for all b222,, nggnnab baab baR. Since G is X-outer, it follows that 20ana. Hence 111aaR. Also since bb for all bR, we have that 1CenaR. Note that for all gG, implies 1g111ag gaag111gaa. So . Thus 1Cena GRGCenCen GRG R . Conversely, is clear.  Cen CenGRR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ggGtr aa, which is called the trac e of a. Also for a right ideal I of R, the right ideal ggGtr IaaI of RG is called the trace of I. Say 1,,nGg g. we put 1ntg gRG. For rR and 112 2nnag agag  RG with iaR, define 111nngggg n. Then R is a right R*G-module. Moreover, we see that rra ra GRGRR is an G,GRR-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 RRIeR and RrI fR with fBR. Then 1ef. Proof. Since R is semiprime, 1essRRRRIlrI fR. Thus 1essRRIfQR . As essRRIeR, essRRIeQ R. We note that e and 1f are in BQR. So we have that . 1efProposition 2.7  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, RrI 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 gII for every gG, where ggIaa 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 pessRRBQReBQRxR RxReR Let be the subring of generated by ˆpqBQRBQRQRR and . pLemma 2.8  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. ˆpqBQR2) R is p.q.-Baer if and only if . pBQR 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 1nIRa RRaR is a finitely generated G-invariant two-sided ideal of R with i. Then aRIG is a two-sided ideal of R*G. Moreover, 1,nIGRGaRGRGa RG  Note that R*G is semiprime by Lemma2.3, So Propo-sition 2.7 yields that there exists such that leSRGessRGRIGeRG G. Since R*G is semipr ime, eBRGG by . Hence by Lemma2.4, . First, we see that CeneRessRRIeR. For this, let 0 with rer eRR. As essRGeRGIG RG, there exists RG such that 0erI G . Say 11 nnbg bg with and ibRigG for 1,i, .n Then 11 nnererb gerbgI G . Hence 0jerb I for some , so jessRRIeR. As G2Cenee R, IeRe, and so esseRe eReIeRe. Now we show that  1RrI eR. If 0e, then RrI R. S o we may a ss ume t ha t . Note that eRe 0eis semiprime and esseRe eReIeRe, and so 0eRerI. Hence 0RReRr IeRe r I. As , IeR1ReRr I. From the modular law, 1RRrIeReRrI. But since 0ReRr I, . There-fore R is G-p.q.-Baer.  1RrI eRConversely, let R be G-p.q.-Baer. Take peBQRG. Then Cen GeQR by Lemma 2.4 since G is also X-outer on as was noted. Also there exists QRRG such that  essRGRRGRGeRGG because QRG is the maximal right ring of quotients of RG (Lemma 2.5) and . Say peBQRG1122nnag agag with and iaRigG for 1, 2,,in. Then eR GeRG R and so iae for each 1, 2,,in. Consider 1,giigGnKRaR. Then K is a finitely generated G-invariant two-sided ideal of R. Further, KeR because . By Cen GeQRessthe preceding argument, we see that RRKeR. From the assumption, there exists lfSRBR such that RrK fR. Thus 1efRG by Lemma 2.6. There- fore eR R, so . From pRGRGBQOpen Access APM B. LI, H. L. JIN Open Access APM 669 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.  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  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  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]). 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