On Regular Rings Satisfying Almost Comparability

In this article, we study regular rings satisfying almost comparability. We first show that, for regular rings, almost comparability is inherited by finitely generated projective modules and finite matrix rings, and, as a main result, we prove that the strict cancellation property holds for the family of all finitely generated projective modules over directly finite regular rings satisfying almost comparability.     

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.     

von-Neumann正则环与左SF-环

环R称为左SF-环,如果每个单左R-模是平坦的.众所周知,Von-Neumann正则环是SF-环,但SF-环是否是正则环至今仍是公开问题,本文主要研究左SF-环是正则环的条件,证明了:如果R是左SF-环且R的每个极大左(右)理想是广义弱理想,那么R是强正则环.并且推广了Rege[3]中的相应结果.     

Von Neumann Regular Rings Satisfying Weak Comparability

The notion of weak comparability was first introduced by K.C. O’Meara, to prove that directly finite simple regular rings satisfying weak comparability must be unit-regular. In this paper, we shall treat (non-necessarily simple) regular rings satisfying weak comparability and give some interesting results. We first show that directly finite regular rings satisfying weak comparability are stably finite. Using the result above, we investigate the strict cancellation property and the strict unperforation property for regular rings satisfying weak comparability, and we show that these rings have the strict unperforation property, which means that nA ≺ nB implies A ≺ B for any finitely generated projective modules A, B and any positive integer n.      

关于右SF-环的正则性

研究了每一个极大的右理想是拟理想的右SF-环的正则性,得到了右SF-环是正则环的一些新的刻画,推广了一些已知的结论.     

关于每一个极大左理想是弱右理想的环

研究了每一个极大左理想是弱右理想的环的性质．得到了SF-环和强正则环的一些新的刻画,推广了一些已知的结论．     

正则环与YJ-内射性

文献 [1]中 ,Ming.R.Y.C引进了 YJ 内射模的概念 ,且指出正则环上的每个模均是 YJ 内射模 ,那么反之如何呢 ?文 [1]中做了一些结果 ,本文拟就这个问题作进一步讨论 .     

On the equations defining monomial curves

Let C be a monomial curve in three dimensional projective space over an algebraically closed field K of characteristic zero . We give several necessary criterions for C to be set theoretic complete intersection. Using these criterions we get several restrictions concerning the form of the equations that define C set theoretically.     

SF-环的内射性

研究了SF-环与P-内射环的关系,构造了SF-环成为P-内射环的一系列条件.证明了SF-环R只要满足其中之一:R的每个极大左理想是有限生成的;特殊右零化子的降链条件;对R的每个极大左理想M,l(M)在R中是本质的,那么R就是P-内射环.在此基础上,利用一定条件下SF-环的P-内射性,发展了SF-环的若干新结果,这些结果部分地拓展了有关文献中的结果.     

关于每一个极大左理想是弱右理想的环(英文)

研究了每一个极大左理想是弱右理想的环的性质.得到了左SF-环和强正则环的一些新的刻画,推广了一些已知的结论.     

On Quotient Rings in Alternative Rings

We introduce a notion of left nonsingularity for alternative rings and prove that an alternative ring is left nonsingular if and only if every essential left ideal is dense, if and only if its maximal left quotient ring is von Neumann regular (a Johnson-like Theorem). Finally, we obtain a Gabriel-like Theorem for alternative rings.     

主左理想由若干个幂等元生成的环

环Ｒ称为左ＰＩ－环，是指Ｒ的每个主左理想由有限个幂等元生成．本文的主要目的是研究左ＰＩ－环的ｖｏｎ Ｎｅｕｍａｎｎ正则性，证明了如下主要结果：（１）环Ｒ是Ａｒｔｉｎ半单的当且仅当Ｒ是正交有限的左ＰＩ－环；（２）环Ｒ是强正则的当且仅当Ｒ是左ＰＩ－环，且对于Ｒ的每个素理想Ｐ，Ｒ／Ｐ是除环；（３）环Ｒ是正则的且Ｒ的每个左本原商环是Ａｒｔｉｎ的当且仅当Ｒ是左ＰＩ－环且Ｒ的每个左本原商环是Ａｒｔｉｎ的；（４）环Ｒ是左自内射正则环且Ｓｏｃ（_ＲＲ）≠０当且仅当Ｒ是左ＰＩ－环且它包含内射极大左理想；（５）环Ｒ是ＭＥＬＴ正则环当且仅当Ｒ是ＭＥＬＴ左ＰＩ－环．     

von Neumann Regular Rings and Right SF-rings

A ring R is called a left (right) SF-ring if all simple left (right) R-modules are flat. It is known that von Neumann regular rings are left and right SF-rings. In this paper, we study the regularity of right SF-rings and prove that if R is a right SF-ring whose all maximal (essential) right ideals are GW-ideals, then R is regular.     

Localization and Catenarity in Strongly Graded Rings

The objective of this article is the study of localization and catenarity in strongly G-graded rings with Noetherian base ring, where G is a finitely generated, nilpotent and torsionfree group. We generalize some results of Guédénon (2000 Guédénon , T. ( 2000 ). Anneaux munis d'une action de groupe superrésoluble . Algebras Groups and Geometries 17 : 17 – 48 . [CSA]  [Google Scholar]). It follows from Corollary 2.6 that if G is free Abelian of finite rank and A is a commutative strongly G-graded ring with base ring a Noetherian regular integral domain, then A is a Noetherian regular integral domain.     

Morphic-环与SF-环

以正则环为桥梁,研究了morphic-环与SF-环之间的关系.主要工作如下:(i)研究了SF-环成为morphic-环的若干条件;(ii)讨论了在一定条件下SF-环与morphic-环的等价性;(iii)给出了利用morphic-环对半单环在约化条件下的一个刻划.     

On Von Neumann Regular Rings of Skew Generalized Power Series

In this paper we introduce a construction called the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S which extends Ribenboim's construction of generalized power series rings. In the case when S is totally ordered or commutative aperiodic, and ω(s) is constant on idempotents for some s ∈ S?{1}, we give sufficient and necessary conditions on R and S such that the ring R[[S, ω]] is von Neumann regular, and we show that the von Neumann regularity of the ring R[[S, ω]] is equivalent to its semisimplicity. We also give a characterization of the strong regularity of the ring R[[S, ω]].     

正则环的投射根

研究了正则环上投射根的性质.证明了正则环的投射根左右对称,且模去投射根的正则环只有零投射根.给出了矩阵环及角落环投射根的计算式,并得到了投射根为零的正则环的一些刻画。最后讨论了投射根为零的正则环在各种环运算下的封闭性和正则环的MP-维数.     

N-环Von-Neumann正则性

环R称为N-环,如果R的素根N(R)=｛r∈R｜存在自然数n使rⁿ=0}.本文不仅对N-环进行了刻划,而且还研究了N-环的VonNeumann正则性.特别证明了：对于N-环R,如下条件是等价的：(1)R是强正则环;(2)R是正则环;(3)R是左SP-环;(4)R是右SF-环;(5)R是MELT,左p-V-环;(6)R是MERT,右p-V-环.因此推广了文献[4]中几乎所有的重要结果,同时也改进或推广了其它某些有关正则环的有用结果.     

Annihilator Ideal-Based Zero-Divisor Graphs Over Multiplication Modules

For a commutative ring R with identity, an ideal-based zero-divisor graph, denoted by Γ _I (R), is the graph whose vertices are {x ∈ R?I | xy ∈ I for some y ∈ R?I}, and two distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we investigate an annihilator ideal-based zero-divisor graph by replacing the ideal I with the annihilator ideal Ann(M) for a multiplication R-module M. Based on the above-mentioned definition, we examine some properties of an R-module over a von Neumann regular ring, and the cardinality of an R-module associated with Γ _Ann(M)(R).