Regular Rings With Generalized s-Comparability

In this paper, we investigate generalized s-comparability over regular rings. These also give generalizations of the corresponding results over regular rings satisfying general comparability.AMS Subject Classification (1991) 16A30 16E50     

Generalized Stable Exchange Rings

In this paper, we investigate diagonal reductions of matrices over generalized stable exchange rings. We show that every regular matirx over generalized stable exchange rings with stable range 2 admits power diagonal reduction.AMS Subject Classification (1991): 16A30 16E50This work was supported by the National Natural Science Foundation of China.     

On Generalized Stable Ideals

An ideal I of a ring R is generalized stable in case aR + bR = R with a ∈ I, b ∈ R implies that there exist s, t ∈ 1 + I such that s(a + by)t = 1 for a y ∈ R. We establish, in this article, necessary and sufficient conditions for an ideal of a regular ring to be generalized stable. It is shown that every regular square matrix over such ideals admits a diagonal reduction. These extend the corresponding results of generalized stable regular rings.     

Abel正则球与完全正则半群

本文分别讨论了关于结合环和半群的二个定理，并且由结合环的这二个定理推出了如下准则：结合环Ｒ是Ａｂｅｌ正则的，当且仅当Ｒ的每个拟理想是正则环．     

Containing Symbolic Powers in Regular Rings

We extend some results of Ein–Lazarsfeld-Smith, Hochster–Huneke, and Takagi-Yoshida concerning the containments of symbolic powers of ideals in ordinary powers in regular rings.     

关于正则环的理想

本文把对于正则环的可比较性推广到对于正则环上理想的可比较性,得到这 种可比较性的若干刻画.进而我们证明,对稳定秩1的理想, Roth问题有肯定解答.     

Characterizations of Strongly Regular Rings

ＣｈａｒａｃｔｅｒｉｚａｔｉｏｎｓｏｆＳｔｒｏｎｇｌｙＲｅｇｕｌａｒＲｉｎｇｓＺｈａｎｇＪｕｌｅ（章聚乐）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＡｎｈｕｉＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｗｕｈｕ２４１０００）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉ...     

Separative Cancellation over Regular Rings

This article develops element-wise characterizations of separative ideals in regular rings. These give connections among separativity, annihilators, cokernels and unit-regularity. For strongly separativity, the analogs are also obtained.     

Regular Action in Rings

Let R be a ring with identity, X(R) the set of all nonzero non-units of R and G(R) the group of all units of R. By considering left and right regular actions of G(R) on X(R), the following are investigated: (1) For a local ring R such that X(R) is a union of n distinct orbits under the left (or right) regular action of G(R) on X(R), if J ⁿ  ≠ 0 = J ⁿ⁺¹ where J is the Jacobson radical of R, then the set of all the distinct ideals of R is exactly {R, J, J ²,…, J ⁿ , 0}, and each orbit under the left regular action is equal to the one under the right regular action. (2) Such a ring R is left (and right) duo ring. (3) For the full matrix ring S of n × n matrices over a commutative ring R, the number of orbits under left regular action of G(S) on X(S) is equal to the number of orbits under right regular action of G(S) on X(S); the result also holds for the ring of n × n upper triangular matrices over R.     

正则环的内射理想

设R为正则环, A∈ M_n(R). 如果M_n(R)AM_n(R)为M_n(R) 的内射理想,该文证明了有U, V∈ GL_n(R)使得UAV为对角矩阵.     

关于m-fold稳定环

本文给出了R为m-fold稳定环的若干充分必要条件，证明了整闭整环的Kronecker函数环m-fold稳定环，进一步地，得到了左(右)拟DUO替换环为m-fold稳定环的条件。     

On Ideals of Regular Rings

In this paper, we investigate ideals of regular rings and give several characterizations for an ideal to satisfy the comparability. In addition it is shown that, if I is a minimal two-sided ideal of a regular ring R, then I satisfies the comparability if and only if I is separative. Furthermore, we prove that, for ideals with stable range one, Roth's problem has an affirmative solution. These extend the corresponding results on unit-regularity and one-sided unit-regularity. Received February 20, 2001, Accepted July 20, 2001     

关于正则模的自同态环

本文研究正则模Ｐ的自同态环Ｓ，主要给出使Ｓ的稳定秩为１的Ｐ的几个充分条件．     

Separativity of Regular Rings in Which 2 Is Invertible

A regular ring R is separative provided that for all finitely generated projective right R-modules A and B, A⊕ A? A⊕ B? A⊕ B implies that A? B. We prove, in this article, that a regular ring R in which 2 is invertible is separative if and only if each a ∈ R satisfying R(1 ? a ²)R = Rr(a) = ?(a)R and i(End _R (aR)) = ∞ is unit-regular if and only if each a ∈ R satisfying R(1 ? a ²)R ∩ RaR = Rr(a) ∩ ?(a)R ∩ RaR and i(End _R (aR)) = ∞ is unit-regular. Further equivalent characterizations of such regular rings are also obtained.     

On the Finiteness of Noetherian Rings with Finitely Many Regular Elements

Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if R \ ZD(R) is finite, then R is finite.     

Regular Rings Satisfying Generalized Almost Comparability

We study regular rings satisfying generalized almost comparability. First, we determine the forms of regular rings satisfying generalized almost comparability. Next, using the above result, we treat the strict cancellation property and the strict unperforation property for regular rings satisfying generalized almost comparability.     

广义Dirichlet级数的正规增长性

该文研究了广义Dirichlet级数在角形域中的增长性及正规增长性, 以及广义Dirichlet级数在角形域中的增长与半实轴的增长之间的关系.     

置换环上的稳定矩阵

本文证明了置换环上的正则稳定矩阵是幂等矩阵和可逆矩阵的积,进一步证明了置换环上的正则稳定矩阵可以对角化。     

Quasipolar Property of Generalized Matrix Rings

The article concerns the question of when a generalized matrix ring K _s (R) over a local ring R is quasipolar. For a commutative local ring R, it is proved that K _s (R) is quasipolar if and only if it is strongly clean. For a general local ring R, some partial answers to the question are obtained. There exist noncommutative local rings R such that K _s (R) is strongly clean, but not quasipolar. Necessary and sufficient conditions for a single matrix of K _s (R) (where R is a commutative local ring) to be quasipolar is obtained. The known results on this subject in [5 Cui , J. , Chen , J. ( 2011 ). When is a 2 × 2 matrix ring over a commutative local ring quasipolar? Comm. Alg. 39 : 3212 – 3221 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] are improved or extended.     

On Prime Rings Satisfying a Certain Generalized Polynomial Identity

Abstract

Recently, Beidar, Fong and Bokut proved that a prime ring satisfies a nontrivial semigroup generalized identity if and only if its central closure is a primitive ring with nonzero socle and the associated skew field is a field. We shall extend their result to the case when generalized polynomial contains three summands.