首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到10条相似文献,搜索用时 156 毫秒
1.
Frank Loose 《代数通讯》2013,41(7):2395-2416
Abstract

A ring R is called left P-injective if for every a ∈ R, aR = r(l(a)) where l? ) and r? ) denote left and right annihilators respectively. The ring R is called left GP-injective if for any 0 ≠ a ∈ R, there exists n > 0 such that a n  ≠ 0 and a n R = r(l(a n )). As a response to an open question on GP -injective rings, an example of a left GP-injective ring which is not left P-injective is given. It is also proved here that a ring R is left FP -injective if and only if every matrix ring 𝕄 n (R) is left GP-injective.  相似文献   

2.
《代数通讯》2013,41(10):5105-5116
Abstract

A ring R is called left IP-injective if every homomorphism from a left ideal of R into R with principal image is given by right multiplication by an element of R. It is shown that R is left IP-injective if and only if R is left P-injective and left GIN (i.e., r(I ∩ K) = r(I) + r(K) for each pair of left ideals I and K of R with I principal). We prove that R is QF if and only if R is right noetherian and left IP-injective if and only if R is left perfect, left GIN and right simple-injective. We also show that, for a right CF left GIN-ring R, R is QF if and only if Soc(R R ) ? Soc( R R). Two examples are given to show that an IP-injective ring need not be self-injective and a right IP-injective ring is not necessarily left IP-injective respectively.  相似文献   

3.
A ring R is called right Johns if R is right noetherian and every right ideal of R is a right annihilator. R is called strongly right Johns if the matrix ring M n (R) is right Johns for each integer n ≥ 1. The Faith–Menal conjecture is an open conjecture on QF rings. It says that every strongly right Johns ring is QF. It is proved that the conjecture is true if every closed left ideal of the ring R is finitely generated. This result improves the known result that the conjecture is true if R is a left CS ring.  相似文献   

4.
Dave Benson  Leonard Evens 《代数通讯》2013,41(10):3447-3451

In this article, we call a ring R right generalized semiregular if for any a ∈ R there exist two left ideals P, L of R such that lr(a) = PL, where P ? Ra and Ra ∩ L is small in R. The class of generalized semiregular rings contains all semiregular rings and all AP-injective rings. Some properties of these rings are studied and some results about semiregular rings and AP-injective rings are extended. In addition, we call a ring R semi-π-regular if for any a ∈ R there exist a positive integer n and e 2 = e ∈ a n R such that (1 ? e)a n  ∈ J(R), the Jacobson radical of R. It is shown that a ring R is semi-π-regular if and only if R/J(R) is π-regular and idempotents can be lifted modulo J(R).  相似文献   

5.
Lixin Mao 《代数通讯》2013,41(12):4319-4327
In this article, we study the weak global dimension of coherent rings in terms of the left FP-injective resolutions of modules. Let R be a left coherent ring and ? ? the class of all FP-injective left R-modules. It is shown that wD(R) ≤ n (n ≥ 1) if and only if every nth ? ?-syzygy of a left R-module is FP-injective; and wD(R) ≤ n (n ≥ 2) if and only if every (n ? 2)th ? ?-syzygy in a minimal ? ?-resolution of a left R-module has an FP-injective cover with the unique mapping property. Some results for the weak global dimension of commutative coherent rings are also given.  相似文献   

6.
A. Majidinya  K. Paykan 《代数通讯》2013,41(12):4722-4750
We say a ring R is (centrally) generalized left annihilator of principal ideal is pure (APP) if the left annihilator ? R (Ra) n is (centrally) right s-unital for every element a ∈ R and some positive integer n. The class of generalized left APP-rings includes generalized left (principally) quasi-Baer rings and left APP-rings (and hence left p.q.-Baer rings, right p.q.-Baer rings, and right PP-rings). The class of centrally generalized left APP-rings is closed under finite direct products, full matrix rings, and Morita invariance. The behavior of the (centrally) generalized left APP condition is investigated with respect to various constructions and extensions, and it is used to generalize many results on generalized PP-rings with IFP and semiprime left APP-rings. Moreover, we extend a theorem of Kist for commutative PP rings to centrally generalized left APP rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. Furthermore, we give a complete characterization of a considerably large family of centrally generalized left APP rings which have a sheaf representation.  相似文献   

7.
In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if and only if (i) for any nonzero right module, there is a nonzero linear map from it to a projective module; (ii) every submodule of RR is not a radical module for some right coherent rings. We call a ring a right X ring if Homa(M, R) = 0 for any right module M implies that M = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover we characterize semisimple rings by using X rings. A famous Faith‘s conjecture is whether a semipimary PF ring is a QF ring. Similarly we study the relationship between X rings and QF and get many interesting results.  相似文献   

8.
ABSTRACT

A ring R is called an n-clean (resp. Σ-clean) ring if every element in R is n-clean (resp. Σ-clean). Clean rings are 1-clean and hence are Σ-clean. An example shows that there exists a 2-clean ring that is not clean. This shows that Σ-clean rings are a proper generalization of clean rings. The group ring ?(p) G with G a cyclic group of order 3 is proved to be Σ-clean. The m× m matrix ring M m (R) over an n-clean ring is n-clean, and the m×m (m>1) matrix ring M m (R) over any ring is Σ-clean. Additionally, rings satisfying a weakly unit 1-stable range were introduced. Rings satisfying weakly unit 1-stable range are left-right symmetric and are generalizations of abelian π-regular rings, abelian clean rings, and rings satisfying unit 1-stable range. A ring R satisfies a weakly unit 1-stable range if and only if whenever a 1 R + ˙˙˙ a m R = dR, with m ≥ 2, a 1,…, a m, d ∈ R, there exist u 1 ∈ U(R) and u 2,…, u m ∈ W(R) such that a 1 u 1 + ? a m u m = Rd.  相似文献   

9.
《代数通讯》2013,41(11):4485-4494
Abstract

Let R be a ring. We prove that every right CF ring is right artinian under the left perfect or strongly right C2 condition. We also show that a right noetherian, left P-injective, left CS-ring is QF.  相似文献   

10.
《代数通讯》2013,41(12):6149-6159
Abstract

A commutative ring R is said to satisfy property (P) if every finitely generated proper ideal of R admits a non-zero annihilator. In this paper we give some necessary and sufficient conditions that a ring satisfies property (P). In particular, we characterize coherent rings, noetherian rings and Π-coherent rings with property (P).  相似文献   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号