共查询到20条相似文献,搜索用时 46 毫秒
1.
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
2.
《代数通讯》2013,41(5):2403-2416
Abstract In this paper, we investigate stable range conditions over extensions of matrix rings. It is shown that a ring R satisfies (s, 2)-stable range if and only if R has a complete orthogonal set {e 1,…, e n } of idempotents such that all e i Re i satisfy (s, 2)-stable range. Also we extend this result to (s, 2)-rings and rings satisfying unit 1-stable range. 相似文献
3.
Exchange rings having ideal-stable range one 总被引:1,自引:0,他引:1
In this paper, we introduce the notion of the ideal-stable range one condition for exchange rings. Some characterizations
for this condition are given. Moreover, we show that, for an exchange ringR, ifI is an ideal ofR andR hasI-stable range one, then every regular square matrix overI is the product of an idempotent matrix and an invertible matrix overR, and admits a diagonal reduction. 相似文献
4.
In this paper, we investigate the comparability structure over exchange rings. It is shown that the subdirect product of
an exchange ring with stable range one and an exchange ring satisfying the comparability is also an exchange ring satisfying
the comparability. This provides a new class of exchange rings satisfying the comparability. Furthermore, we investigate the
s-comparability over exchange rings. This generalizes the corresponding results of Goodearl and Chen.
Received November 26, 1999, Accepted February 5, 2001 相似文献
5.
《代数通讯》2013,41(7):3089-3098
This paper studies exchange rings R such that R/J(R) has bounded index of nilpotence. We give several characterizations of such rings. We prove that if a semiprimitive exchange ring R has index n, then for any maximal two-sided I of R, if R/I has length n, then there exists a central idempotent element e in R such that eRe is an n by n full matrix ring over some exchange ring with central idempotents, and the restriction π from eRe to R/I is surjective. 相似文献
6.
Huanyin Chen 《Czechoslovak Mathematical Journal》2008,58(4):899-910
Let R be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in R is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related
comparability. 相似文献
7.
Víctor Jiménez López 《代数通讯》2013,41(1):63-76
Hirano studied the quasi-Armendariz property of rings, and then this concept was generalized by some authors, defining quasi-Armendariz property for skew polynomial rings and monoid rings. In this article, we consider unified approach to the quasi-Armendariz property of skew power series rings and skew polynomial rings by considering the quasi-Armendariz condition in mixed extension ring [R; I][x; σ], introducing the concept of so-called (σ, I)-quasi Armendariz ring, where R is an associative ring equipped with an endomorphism σ and I is an σ-stable ideal of R. We study the ring-theoretical properties of (σ, I)-quasi Armendariz rings, and we obtain various necessary or sufficient conditions for a ring to be (σ, I)-quasi Armendariz. Constructing various examples, we classify how the (σ, I)-quasi Armendariz property behaves under various ring extensions. Furthermore, we show that a number of interesting properties of an (σ, I)-quasi Armendariz ring R such as reflexive and quasi-Baer property transfer to its mixed extension ring and vice versa. In this way, we extend the well-known results about quasi-Armendariz property in ordinary polynomial rings and skew polynomial rings for this class of mixed extensions. We pay also a particular attention to quasi-Gaussian rings. 相似文献
8.
《代数通讯》2013,41(12):5477-5497
A ring R is called left T-idempotent (left T-stable) if for any sequence r 1,r 2,… in R there exists a positive integer n such that r 1 r 2… rn is an idempotent (r 1 r 2… rn = r 1 r 2… r n+1 = …). In this paper we characterize left T-idempotent rings and left T-stable rings and describe left T-idempotent semigroup rings and left T-stable semigroup rings. 相似文献
9.
Huanyin Chen 《Linear and Multilinear Algebra》2013,61(1):81-92
Let I be an ideal of a ring R. We say that R is a generalized I-stable ring provided that aR+bR=R with a?∈?1+I,b?∈?R implies that there exists a y?∈?R such that a+by?∈?K(R), where K(R)={x?∈?R?∣?? s, t?∈?R such that sxt=1}. Let R be a generalized I-stable ring. Then every A?∈?GLn (I) is the product of 13n?12 simple matrices. Furthermore, we prove that A is the product of n simple matrices if I has stable rank one. This generalizes the results of Vaserstein and Wheland on rings having stable rank one. 相似文献
10.
Huanyin Chen 《代数通讯》2013,41(8):3913-3924
In this paper, we show that a ring R satisfies unit 1-stable range if and only if a1R + ? + amR = dR with m ≥ 2,a 1, ?am ?R implies that there exist u1 , ?um ? U(R) such that a1u1 +?+amum = d and an exchange ring R has stable range one if and only if a1R+?+amR = dR with m ≥ 2,a 1,?,am ? R implies that there exist unit-regular w 1,?,wm ? R such that a1w1 +?+ amwm = d. Also we show that an exchange ring R satisfies the n-stable range condition if and only if a( nR)+bR = dR with a ? Rn,b,d ? R implies that there exist unimodular regular w ? n R and: y ? R such that aw+by = d. 相似文献
11.
Huanyin Chen 《代数通讯》2013,41(11):5223-5233
In this paper,we investigate power-substitution over exchange rings.We show that an exchange ring R satisfies power-substitution if and only if for any regular x ∈ R, there exists a positive integer n such that xI n is unit πregular in M n(R). 相似文献
12.
LiQiongXU WeiMinXUE 《数学学报(英文版)》2003,19(1):141-146
Let n be an integer with |n| > 1. If p is the smallest prime factor of |n|, we prove that a minimal non-commutative n-insertive ring contains n
4 elements and these rings have five (2p+4) isomorphic classes for p = 2 (p ≠ 2).
This research is supported by the National Natural Science Foundation of China, and the Scientific Research Foundation for
“Bai-Qian-Wan” Project, Fujian Province of China 相似文献
13.
Huanyin Chen 《Southeast Asian Bulletin of Mathematics》2001,25(2):209-216
In this paper, we show that if rings A and B are (s, 2)-rings, then so is the ring of a Morita context (A, B, M, N, , ). Also we get analogous results for unit 1-stable ranges and GM-rings. These give new classes of rings satisfying such stable range conditions.2000 Mathematics Subject Classification: 16U99 16E50 相似文献
14.
15.
Harpreet K. Grover 《代数通讯》2013,41(9):3288-3305
A ring R is said to be von Newmann local (VNL) if for any a ∈ R, either a or 1 ?a is (von Neumann) regular. The class of VNL rings lies properly between exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize and classify arbitrary VNL rings without an infinite set of orthogonal idempotents; and also the VNL rings having a primitive idempotent e such that eRe is not a division ring. We prove that a semiperfect ring R is VNL if and only if for any right uni-modular row (a 1, a 2) ∈ R 2, one of the a i 's is regular in R. Formal triangular matrix rings that are VNL are also characterized. As a corollary, it is shown that an upper triangular matrix ring T n (R) is VNL if and only if n = 2 or 3 and R is a division ring. 相似文献
16.
《代数通讯》2013,41(2):907-925
In this paper we characterize the largest exchange ideal of a ring R as the set of those elements x ∈ R such that the local ring of R at x is an exchange ring. We use this result to prove that if R and S are two rings for which there is a quasi-acceptable Morita context, then R is an exchange ring if and only if S is an exchange ring, extending an analogue result given previously by Ara and the second and third authors for idempotent rings. We introduce the notion of exchange associative pair and obtain some results connecting the exchange property and the possibility of lifting idempotents modulo left ideals. In particular we obtain that in any exchange ring, orthogonal von Neumann regular elements can be lifted modulo any one-sided ideal. 相似文献
17.
Abstract In this paper we introduce generalized ideal-stable regular rings. It is shown that if a regular ring R is a generalized I-stable ring, then every square matrix over I is the product of an idempotent matrix and an generalized invertible matrix and admits a diagonal reduction by some generalized invertible matrices. 相似文献
18.
An associative ring R with unit element is called semilocal if R modulo its Jacobson radical is an artinian ring. It is proved that the multiplicative group R* of a semilocal ring R generated by R* satisfies an n-Engel condition for some positive integer n if and only if R is m-Engel as a Lie ring for some positive integer m depending only on n.Received: 21 January 2003 相似文献
19.
We are interested in (right) modules M satisfying the following weak divisibility condition: If R is the underlying ring, then for every r ∈ R either Mr = 0 or Mr = M. Over a commutative ring, this is equivalent to say that M is connected with regular generics. Over arbitrary rings, modules which are “minimal” in several model theoretic senses satisfy this condition. In this article, we investigate modules with this weak divisibility property over Dedekind-like rings and over other related classes of rings. 相似文献
20.
《代数通讯》2013,41(6):2589-2595
It is shown that if e is an idempotent in a ring R such that both eRe and (1 ? e)R(1 ? e) are clean rings, then R is a clean ring. This implies that the matrix ring M n (R) over a clean ring is clean, and it gives a quick proof that every semiperfect is clean. Other extensions of clean rings are studied, including group rings. 相似文献