馮啟順,儲(chǔ)茂權(quán)

(安徽師范大學(xué)數(shù)學(xué)計(jì)算機(jī)科學(xué)學(xué)院,安徽 蕪湖 241000)

擬G-morphic環(huán)的一些結(jié)果

馮啟順,儲(chǔ)茂權(quán)

(安徽師范大學(xué)數(shù)學(xué)計(jì)算機(jī)科學(xué)學(xué)院,安徽 蕪湖 241000)

在擬morphic環(huán)和G-morphic環(huán)的基礎(chǔ)上,給出了新環(huán)擬G-morphic環(huán)的定義.主要證明了如下結(jié)果:對(duì)交換環(huán)R中任意冪等元e,若R是左擬G-morphic環(huán),則eRe也是左擬G-morphic環(huán);左擬morphic(或左擬G-morphic)的Bear環(huán)是正則環(huán)(或π-正則環(huán));每一個(gè)左擬G-morphic環(huán)都是右GP-內(nèi)射環(huán).

擬morphic環(huán);擬G-morphic環(huán);Bear環(huán);正則環(huán);π-正則環(huán)

1 引言

本文中的環(huán)R均指有單位元的結(jié)合環(huán),環(huán)上的模均指單式模.U(R),E(R)分別表示R的可逆元集,冪等元集.l(a)和r(a)分別表示a在R中的左零化子和右零化子.環(huán)R稱為正則環(huán)[1],若對(duì)于任意a∈R,存在r∈R,使得a=ara;環(huán)R稱為π-正則環(huán)[2],若對(duì)于任意a∈R,存在r∈R,n∈Z+,使得an=anran.顯然,正則環(huán)一定是π-正則環(huán),反之不成立.

2004年,文獻(xiàn) [3]中提出了 morphic環(huán)的定義,而后,文獻(xiàn) [4-5]提出了 G-morphic與擬morphic環(huán)的定義.

若環(huán)R中的元素都是左(右)G-morphic的,則稱環(huán)R為左(右)G-morphic環(huán).

定義 1.3[5]環(huán) R中的元素 a稱為左擬 morphic的,是指 ?b,c∈R,使得 Ra=l(b)且l(a)=Rc.若環(huán)R中的元素都是左擬morphic的,則稱環(huán)R為左擬morphic環(huán).類似可定義右擬morphic環(huán).

2 主要結(jié)果

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[2]Zhang Jule,Wu Jun.Characterizations of P-injective[J].Algebra Colloquium,1999,6(3):277-282.

[3]Nicholson W K,Campos E S.Rings with the dual of the isomorphism theorem[J].J Algebra,2004,207:391-406.

[4]凌燈榮,夏徐林,戴澤儉.G-morphic環(huán)的一些結(jié)果[J].安徽師范大學(xué)學(xué)報(bào),2005,28(4):398-400.

[5]Camillo V,Nicholson W K.Quasi-morphic rings[J].J Algebra Applications,2007,6(5):789-799.

[6]Yue Chi Ming R.On V-rings and prime rings[J].J Algebra,1980,62:13-20.

[7]Xue Weiming.On PP rings[J].Kobe J.Math.,1990,7:77-80.

[8]凌燈榮,儲(chǔ)茂權(quán).Morphic環(huán)和G-morphic環(huán)的一些結(jié)果[J].大學(xué)數(shù)學(xué),2008,24(1):36-38.

[9]Camillo V,Nicholson W K,Wang Z.Left quasi-morphic rings[J].J.Algebra.Applications,2008,7(6):725-733.

[10]Yue Chi ming R.On annihilator ideals IV[J].Riv.Mat.Univ.Parma.,1987,13:19-27.

Some results of quasi-G-morphic rings

Feng Qishun,Chu Maoquan
(College of Mathematics and Computer Science,Anhui Normal University,Wuhu 241000,China)

Based on the quasi-morphic ring and G-morphic ring,we gave the de fi nition of quasi-G-morphic ring. The main results of this paper as follows:If R is a commutative left quasi-G-morphic ring,the same is true of eRe for every idempotent e∈R;Left quasi-morphic(or left quasi-G-morphic)Bear ring is regular ring(or π-regular ring);Every left quasi-G-morphic ring is right GP-injective ring.

quasi-morphic ring,quasi-G-morphic ring,Bear ring,regular ring,π-regular ring

O153.3

A

1008-5513(2012)05-0687-05

2011-12-23.

安徽省教育廳自然科學(xué)研究重點(diǎn)項(xiàng)目(KJ2010A126).

馮啟順(1985-),碩士生,研究方向：環(huán)模理論.

2010 MSC:16W99