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．     

Generalized Stable Regular Rings

Abstract

In this paper we show that a regular ring R is a generalized stable ring if and only if for every x ∈ R, there exist a w ∈ K(R) and a group G in R such that wx ∈ G. Also we show that if R is a generalized stable regular ring, then for any A ∈ M _n (R), there exist right invertible matrices U ₁, U ₂ ∈ M _n (R) and left invertible matrices V ₁, V ₂ ∈ M _n (R) such that U ₁ V ₁ AU ₂ V ₂ = diag(e ₁,…, e _n ) for some idempotents e ₁,…, e _n  ∈ R.     

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.     

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.     

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.     

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.     

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．     

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     

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 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     

Endomorphism Rings of Regular Modules over Wild Hereditary Algebras

Let X be an indecomposable regular module over a connected wild hereditary path-algebra. The main result is a factorization property for maps in the radical of End _H (X) and an upper bound for its degree of nilpotency. The bound is sharp if X has elementary quasi-top. In this, case the socle of End _H (X) can also be characterized.     

Characterizations of Strongly Regular Rings

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

Some Characterizations of Dedekind Rings

In this article we establish some characterizations for Dedekind rings.     

Ideal-comparability over Regular Rings

We introduce the concept of ideal-comparability condition for regular rings. Let I be an ideal of a regular ring R. If R satisfies the I-comparability condition, then R is one-sided unit-regular if and only if so is R/I. Also, we show that a regular ring R satisfies the general comparability if and only if the following hold： （1） R/I satisfies the general comparability; （2） R satisfies the general I-comparability condition; （3） The natural map B（R） → B（R/I） is surjective.     

形式三角矩阵环的可分性和稳定度

In this paper we study the formal triangular matrix ring T =and give some necessary and sufficient conditions for T to be (strongly) separative, m-fold stable and unit 1-stable. Moreover, a condition for finitely generated projec-tive T-modules to have n in the stable range is given under the assumption that A and B are exchange rings.     

正规几乎PP环

首先探讨正规几乎PP环的内刻划及左右对称性；其次，研究正规几乎PP环与PP环的关系，最后证明多项式环R［X」是正规几乎PP环当且仅当R是正规几乎PP环．     

Regular Elements in Quasi-Morphic Rings

We introduce, in this article, a partial order on quasi-morphic rings and investigate the behavior of regular elements in such rings. Further, the cleanness of regular elements in quasi-morphic rings having internal-cancelltion has been derived.     

关于单位正则环

本文得到了单位正则环的一个新特征,证明了:正则环R为单位正则环当且仅当存在理想I使得(1)R/I为单位正则环;(2)对任何a∈R,存在理想J满足JI=0和a=aua,其中u模J左可逆.作为应用,利用零化子理想刻画了单位正则环.