The p.q.-Baer Property of Skew Group Rings under Finite Group Action*

doi:10.4236/apm.2013.38089

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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: #hljin98@ybu.edu.cn, hljin98@hanmail.net

Received October 14, 2013; revised November 14, 2013; accepted November 20, 2013

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.,







lX)

denote a right (resp.,left) annihilator of

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

R, 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 [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.,



ae eae

ea eae

aR







SR)

denote the set of all left (resp., right) semicentral idem-

potents. An idempotent e of a ring R is called semicentral

reduced if









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

SR

If R is a semiprime ring and I is a two-sided ideal of R,

then









lIrI . For a right R-module M and a sub-

module N of M, we use ess

R and

NMden

NM

to denote that NR is essential in MR and NR is dense in MR,

respectively.

Let R is a ring,





Aut R

d enote a group of ring auto-

morphisms of R, G be a subgroup of .



Aut R





The skew group ring R*G is defined to be





,ab R



with addition given component wise and multiplication

given as follows: if



and ,

hG, then









ab bha

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













 with a field F of characteristic 2,

then R is right p.q.-Baer. De fi ne



Aut

R by

11 11

00100

ab ab

gcc







 



 



 

*Project supported by the National Natural Science Foundation o

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





RG of R*G generated by

. By computation, we have

 



000 0

rgRG

xy xxy

xy F





 







 



 











Suppose that

 



rgRGeR



for some . Note that the idempotents of

R*G are 0, 1.

ee RG

00 00









, 00

01 00



 





 



 

with . Since

,ab F







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

 



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

R, let







for each

QRxr rxrR



 ,

where is the Martindale right ring of quotients

of R (see [10] for more on ). We say that



X-outer if 0



. A subgroup G of is called

X-outer on R if every 1



Aut R

G is X-outer. Assume

that R is a semiprime ring, then for



Aut



R, let







for each

QR xrrxrR .

For



Aut

R, we claim that



 . Obviously



 . Conversely, if

x then

RRx. There

exists den

R such that

IR. Therefore ,

RI R



den

RI R, and

RI RxIR. Thus





QR,

hence



. Theref or e



 . 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. 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

a0na 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) [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



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

aag agRen



  with i



1 the identity of G, and i



The





122 122

nn nn

aag agbbaag ag 



for all bR



. So 11

ab ba



for all b

,, n

gnn

ab baab ba



. Since G is X-outer, it follows that

20an



. Hence 11

1aaR



. Also since





for all bR



, we have that





1CenaR.

Note that for all



, implies

g



111

ag gaag



. So . Thus



1Cena



 G





CenCen G

RG R .

Conversely, is clear.

 

Cen Cen

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



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



tr aa







which is called the trac e of a. Also for a right ideal I of R,

the right ideal









tr IaaI









of RG is called

the trace of I. Say





1,,

Gg g. we put

tg g



. For rR



and 112 2nn

ag agag



 



 with i



, define 11

1nn

gg n.

Then R is a right R*G-module. Moreover, we see that

rra ra



 



is an







-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

eR and





rI fR with



BR. Then

1ef



.

Proof. Since R is semiprime,











ess

RRR

lrI fR.

Thus





ess

fQR .

As ess

eR,



ess

eQ R. We note that e and



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,





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

I for every

G, where



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

ess

BQR

eBQRxR RxReR

Let be the subring of generated by



ˆpqB



BQR



R and .





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

2) R is p.q.-Baer if and only if .



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

Ra RRaR

is a finitely generated G-invariant two-sided ideal of R

with i. Then aR

G is a two-sided ideal of R*G.

Moreover,



GRGaRGRGa RG 

Note that R*G is semiprime by Lemma2.3, So Propo-

sition 2.7 yields that there exists such

that



eSRG







ess



IGeRG G





Since R*G is semipr ime,





eBRG

by [9]. Hence

by Lemma2.4, . First, we see that



CeneR

ess

eR. For this, let 0 with r

er eRR



. As









ess

eRGIG RG



, there exists RG



 such

that 0erI G





 .

Say 11 nn

bg bg





 with and

bRi



for

1,i, .n



 Then







11 nn

ererb gerbgI G





 .

Hence 0j

erb I



 for some , so jess

eR. As



2Cenee R,

eRe, and so ess

eRe eRe

eRe.

Now we show that



 

rI eR. If 0e



, then





rI R



. S o we may a ss ume t ha t . Note that eRe 0e

is semiprime and ess

eRe eRe

eRe, and so





eRe



Hence





eRr IeRe r I





.

As ,

IeR





eRr I



. From the modular law,











rIeReRrI



But since





eRr I



, . There-

fore R is G-p.q.-Baer.

 

rI eR

Conversely, let R be G-p.q.-Baer. Take







eBQRG



.

Then



Cen G

eQR











by Lemma 2.4 since G is also X-outer on as was

noted. Also there exists







 such that

 

ess

RGRGeRG







because





QRG



is the maximal right ring of quotients

of RG



(Lemma 2.5) and . Say





eBQRG





1122nn

ag agag





 with and

aRi



for

1, 2,,in



. Then





eR GeRG







 R and so i



for each 1, 2,,in



. Consider 1,

RaR



. Then

K is a finitely generated G-invariant two-sided ideal of

R. Further,

eR because . By



Cen G

eQR





ess

the preceding argument, we see that

eR. From

the assumption, there exists







SRBR such that





rK fR. Thus 1efR





by Lemma 2.6. There-

fore eR R



, so . From



pRGRGBQ

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.

[10] T. Y. Lam, “Lectures on Modules and Rings,” Springer,

Berlin, 1998.

[11] J. W. Fisher and S. Montgomery, “Semiprime Skew Group

Rings,” Journal of Algebra, Vol. 52, No. 1, 1978, pp.

241-247. http://dx.doi.org/10.1016/0021-8693(78)9027 2-7

[12] S. Montgomery, “Outer Automorphisms of Semi-Prime

Rings,” Journal London Mathematical Society, Vol. 18,

No. 2, 1978, pp. 209-220.

http://dx.doi.org/10.1112/jlms/s2-18.2.209

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