关于半交换环与强正则环

本文得到了环Ｒ是强正则环的若干充分必要条件，证明了下面条件是等价的：（１）Ｒ是强正则的；（２）Ｒ是半交换正则的；（３）Ｒ是半交换的左ＳＦ－环；（４）Ｒ是半交换的ＥＬＴ环，且使得每个单左Ｒ－模是Ｐ－内射的或者平坦的；（５）Ｒ是半交换右非奇异的左ＳＦ－环；（６）Ｒ是半素的半交换左（或右）Ｐ－内射环．     

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．     

每个本质左理想是幂等的MERT环

环Ｒ称为ＭＥＲＴ环，如果Ｒ的每个极大本质右理想是理想．本文证明了：每个本质左理想是幂等的半素ＭＥＲＴ环一定是ｖｏｎＮｅｕｍａｎｎ正则的．于是肯定地回答了Ｍｉｎｇ的一个公开问题．     

F－V－环的广义内射性刻划

设Ｆ是含单位元的结合环Ｒ上的左Ｇａｂｒｉｅｌ拓朴，称Ｒ是Ｆ－Ｖ－环，如果商范畴（Ｒ，Ｆ）－Ｍｏｄ中的所有单对象都是内射对象。本文我们利用左Ｒ－模的ｖＮ－内射性及拟内射性给出Ｆ－Ｖ－环的特征刻划。     

F－V－环的广义内射性刻划

设Ｆ是含单位元的结合环Ｒ上的左Ｇａｂｒｉｅｌ拓朴，称Ｒ是Ｆ－Ｖ－环，如果商范畴（Ｒ，Ｆ）－Ｍｏｄ中的所有单对象都是内射对象。本文我们利用左Ｒ－模的ｖＮ－内射性及拟内射性给出Ｆ－Ｖ－环的特征刻划。     

正则环与YJ—内射性

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？「１」中做了一些结果,本拟就这个问题作进一步讨论。     

关于环的极大本质右理想

设Ｒ为环，我们考虑下面两个条件。（＊）Ｒ的每个极大本质右理想是ＧＰ－内射右Ｒ－模或右零化子．（＊）Ｒ的每个极大本质右理想是ＹＪ－内射右Ｒ－模．本文旨在研究满足条件（＊）或（＊）的环，同时我们还给出了强正则环和除环的一些新刻画．     

无挠左（右）Artin环是拟Frobenius环

无挠左（右）Ａｒｔｉｎ环是拟Ｆｒｏｂｅｎｉｕｓ环乌成伟（吉林工学院基础部，长春１３００１２）关键词内积，左（右）内零化子，自内射环．分类号ＡＭＳ（１９９１）１６Ｄ５０／ＣＣＬＯ１５３．３设Ｒ为有１的左（右）Ａｒｔｉｎ环，如果对于任一整数洲与ｒ∈Ｒ，ｍ...     

关于SF－环的几点注记

本文中，我们证明了如下主要结果：Ⅰ　对于环Ｒ，下面条件是等价的：（１）Ｒ是Ａｒｔｉｎ半单环；（２）Ｒ是左ＳＦ－环，且Ｒ满足特殊右零化于降链条件；（３）Ｒ是左ＳＦ－环和Ｉ－环，且Ｒ￣Ｒ具有有限Ｇｏｌｄｉｅ维数。Ⅱ对于环Ｒ，下面条件是等价的：（１）Ｒ是ＶｏｎＮｅｕｍａｎｎ正则环；（２）Ｒ是左ＳＦ－环，且每个苛异循环左Ｒ－模的极大子模是平坦的。     

关于分次环的分次性质和非分次性质

本文证明了对有限群分次环Ｒ而言,下列条件等价：（１）Ｒ是左ｇｒ－自内射环（左ｇｒ－ＰＦ环,左ｇｒ－ＱＦ环,左ｇｒ－线性紧环）．（２）Ｒ是左自内射环（左ＰＦ环,左ＱＦ环,左线性紧环）．（３）Ｒ＃Ｇ^*是左自内射环（左ＰＦ环,左ＱＦ环,左线性紧环）．     

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.     

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.     

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.     

von Neumann正则环与右SF-环

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.     

Strongly Regular Rings and SF-rings

in this paper, new characteristic properties of strongly regular rings are' given.Relations between certain generalizations of duo rings are also considered. The followingconditions are shown to be equivalent: (1) R is a strongly regular ring; (2) R is a left SFring such that every product of two independent closed left ideals of R is zero; (3) R is aright SF-ring such that every product of two independent closed left ideals of R is zero; (4)R is a left SF-ring whose every special left annihilator is a quasi-ideal; (5) R is a right SFring whose every special left annihilator is a quasi-ideal; (6) R is a left SF-ring whose everymaximal left ideal is a quasi-ideal; (7) R is a right SF-ring whose every maximal left ideal isa quasi-ideal; (8) R is a left SF-ring such that the set N(R) of all nilpotent elements of R isa quasi-ideal; (9) R is a right SF-ring such that N(R) is a quasi-ideal.     

Left SF-Rings and Regular 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 article, we study the regularity of left SF-rings and we prove the following: 1) if R is a left SF-ring whose all complement left (right) ideals are W-ideals, then R is strongly regular; 2) if R is a left SF-ring whose all maximal essential right ideals are GW-ideals, then R is regular.     

The Relative Properties of Graded Ring R and Smash Product R # G

ＴｈｅＲｅｌａｔｉｖｅＰｒｏｐｅｒｔｉｅｓｏｆＧｒａｄｅｄＲｉｎｇＲａｎｄ　ＳｍａｓｈＰｒｏｄｕｃｔＲ＃ＧＷｅｉＪｕｎｃｈａｏ（魏俊潮）；ＬｉＬｉｂｉｎ（李立斌）（ＹａｎｇｚｈｏｕＩｎｓｔｉｔｕｔｅｏｆＴｅｃｈｎｏｌｏｇｙ，Ｙａｎｇｚｈｏｕ，２２５０...     

Von neumaivn regularity of sf-rings

A ring R is called left (right) SF-ring if all simple left (right) R-modules are flat. It is proved that R is Von Neumann regular if R is a right SF-ring whoe maximal essential right ideals are ideals. This gives the positive answer to a qestion proposed by R. Yue Chi MIng in 1985, and a counterexample is given to settle the follwoing question in the negative: If R is an ERT ring which is one-sided V-ring, is R a left and right V-ring? Some other conditions are given for a SF-ring to be regular.     

Characterizations of Strongly Regular Rings

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

右对称环

本文在左对称环的基础上提出了右对称环的概念,分别给出了是右对称环但不是左对称环和是左对称环但不是右对称环的例子．证明了（1）如果R是Armendariz环,则R是右对称环的充要条件R[x]是右对称环;（2）如果R是约化环,则R[x]/（x^n）是右对称环,其中（xn）是由xn生成的理想．