《关于每一个极大左理想是弱右理想的环》由会员分享，可在线阅读，更多相关《关于每一个极大左理想是弱右理想的环（4页珍藏版）》请在金锄头文库上搜索。
1、第26卷第1期 2010年2月 大 学 数 学 CoLI EGE MATHEMATICS Vol_26，1 Feb2010 On Rings Whose Every Maximal Left Ideal is a Weakly Right Ideal ZHAO Liang ， (Faculty of Science，Jiangxi University of XIONG Xiao-feng Science and Technology，Ganzhou 34 1 000，China) Abstract：Pr。perties。f the rings wh。se eVery maximal left
2、 ideal is a weakly right ideal are investigated s。m。 ew characterizati。ns。n left SFrings and str。ngly regu1ar rings are。btained，which extend severa1 known results Key words：weakly ideals；regular rings；SF-rings CLC Number：O1533 Document Code：A Article ID：1 6721454(2010)010033一O4 1 Intr0ducti0n l hrou
3、ghout th s paperall rings are associative with identity and all modules are unitarvFor any nonempry subset义of a ring R，we denote by r尺(X)and (X)the right and left annihilator of X in R respectively，ie，rR(X)一zR JXx一0and zR(X)一 R X一0 A ing R is calied a left(right)quasiduo ring if every maximal left(r
4、ight)idea1 of R is an idea1 In L l J，the regularity of right quasiduo rings was considered， and severa1 properties were a1so 。b i“ dRecall that a left Rmodule M is called left GPinjective2 if for any 04-口E R，there exists a Pos t Ve mteger刀such that O and any Rhomomorphism from Ra into M extends to o
5、ne from R into MA ring R without nonzero nilpotent elements is called reducedIf R is a reduced ring，then it is clear that r()一 (以)for anyR A o d ng to L3J，an additive subgroup of R is called a weakly left idea1 of R if fOr everv zf and r ，there ex sts a natural number such that(rz) JWeakly right ide
6、als can be defined in a s m l dr wayIt lS obviOUS that every left(right)ideal of R is weakly left(right)idea1。But there is an examp e to show hat the converse is not true in general as indicated in 3， Example 1 We cnaracter ze the regularity of rings whose every maximal left ideal is a weakly right
7、idea1 in this paper Some results are generalized 2 Main Results It is well known that if R is quasiduo ring R is a reduced ringThis result can be improved Lemma 21 Let R be a ring whose every simple left Rmodule is GPinj ective，then R is a and every simple left R as follows module is GPinjective， ma
8、ximal left ideal is a weakly right idea1If every reduced ring Received date：20070606；Revised date：20080225 F。undation item：The Youth Fund of Jiangxi Education Provincia1 Department(GJJ 1 0 1 5 5) 34 大 学 数 学 第26卷 Proof If there exists O口R and a 一0，then Ra+l(a)RIf not，RRa+Z(口)一RaThen 1=ra for some rR，
9、so a=Fa =0，a contradictionThen there exists a maximal left ideal M containing Ra+Z(口)，so RM is】eft GP-injectiveLet 厂：尺以 尺M：r一，+M It is obvious that厂is a welldefined R-homomorphismSince RM is ieft GPinjective and a 0， there exists cER such that 1+Mf(a)=ac+M，so 1-acEMBy assumption。Mis a weakly right i
10、dea1 of R，so aM implies that there exists a natural number such that(ac)”MLet z一1-acE M，then(nc) 一(1一 )”= c (一1) 一 一it follows from M，(c)nMthat 1EM，where k=0 k：=1， ，which contradicts MRTherefore R is a reduced ring Recall that a ring R is strongly regular ringif for any a E R there exists bR satisfi
11、es a=ab A ring R is a left Vring if every simple left Rmodule is injectiveIn view of the preceding lemma，we obtain the following basic equivalences for the regularity of a ring whose every maximal left ideal is a weakly right idea1 Theorem 21 If R is a ring whose every maximal left ideal is a weakly
12、 right ideal，then the following statements are equivalent： (i)Every left Rmodule is GPinjective (ii)Every cyclic left Rmodule is GPinj ective (iii)Every simple left Rmodule is GPinjective (iv)R is a strongly regular ring (v)R is a von Neumann regular ring (vi)R is a left Vring Proof The result follo
13、ws from the proof of Lemma 21 and that of r2，Theorem 1O Let R be a ring and G be a group(not necessarily finite)It is welt known that the group ring REG3 is yon Neumann regular if and only if：(a)R is a von Neumann regular ring(b)G is locally finite(ie，every finitely generated subgroup is also finite
14、)(c)the order of every element of G is a unit in R(see，for example，r5，P155) Corollary 21 Let R be a ring whose every maximal left ideal is a weakly right idea1If G is a group，then the following hold： (i)If REGis a left Vring，then RGis a yon Neumann regular ring (ii)If G is a finite group，then REGis
15、a left Vring if and only if RGis a von Neumann regular ring Proof(i)By E6，Theorem lo2，If RIGis a left Vring，then R is a left V ring，G is a locally finite and the order of each element of G is a unit in RSo RGis a von Neumann regular ring by theorem 21 and the fact mentioned above corollary 21 (ii)By
16、6，Theorem 11，If G is a finite group，then GERis a left Vring if and only if R is a left Vring，G is a locally finite and the order of every element of G is a unit in RAgain the conclusion follows from theorem 21 Recall that a ring R is called right(1eft)weakly regular if for every elementz E R，x E xRxR ( E RxRx)R is called weakly regu
