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.     

FP—内射环和IF环的几个特征

本文给出了FP—内射环和IF环的如下几个特征:(l)R为右FP—内射环当且仅当任意左R—模正合列Kⁿ→Kⁿ→N→0 N为无挠模,当且仅当任一n阶矩阵环为右P—内射环;(2)R为左IF环当且仅当任一有限生成左R—模均可嵌入平坦模;(3)R为IF环当且仅当R为伪凝聚的上平坦环。     

关于AP-内射环的一个注记

本文的主要目的是讨论AP-内射环中的两个问题:(1)环R是正则的当且仅当R是左AP-内射的左PP-环;(2)如果R是左AP-内射环,那么R是内射环当且仅当R是弱内射环.因此我们推广了内射环的一些结果,与此同时我们还取得了一些新的结果.     

Rees环的入射维数

设 A 是左、右 Noether 环,x 是 A 的中心正则元.A_x 表示 A 关于乘闭子集{1,x,x²,…}的局部化.M 是 A-模且 x 是 M 的非零因子.本文确定了入射维数Id_A(M),Id_Ax(M_x)与 Id_A/xA(M/xM)三者之间的等式关系,并把结果应用于滤环(filtered ring)的 Rees 环,得到了 Ekstr(?)m 的两个结果的统一形式和改进,同时推广了 Li Huishi,M.Van den Bergh 和 F.Van Oystaeyen 的相应结果.     

零可换环的一些性质

本文刻画了零可换环的一些性质,同时将交换环上的一些结果推广到零可换环上.对于零可换环R 证明了: (1)R是强正则环当且仅当R中每个为零化子的本质左理想是左GP.内射模或R中存在一个极大左理想K,使得K中每个元索的零化子是左GP-内射模; (2)R是GPP-环当且仅当R是拟π-正则的GPF-环.     

零可换环的一些性质

本文刻画了零可换环的一些性质,同时将交换环上的一些结果推广到零可换环上.对于零可换环R证明了(1)R是强正则环当且仅当R中每个为零化子的本质左理想是左GP-内射模或R中存在一个极大左理想K,使得K中每个元素的零化子是左GP-内射模;(2)R是GPP-环当且仅当R是拟π-正则的GPF-环.     

拟G-morphic环的一些结果

在拟morphic环和G-morphic环的基础上,给出了新环拟G-morphic环的定义.主要证明了如下结果:对交换环R中任意幂等元e,若R是左拟G-morphic环,则eRe也是左拟G-morphic环;左拟morphic(或左拟G-morphic)的Bear环是正则环(或π-正则环);每一个左拟G-morphic环都是右GP-内射环.     

Von Neumann正则环和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-环是否是正则环的问题至今仍是公开的.本文研究左 SF-环是正则环的条件,证明了,如果下列之一成立,那么左 SF-环是正则的:(1)循环模的每个极大子模是平坦的;(2)不可分解的商环是左 quasi-duo;(3)极大左理想的左零化子是本质的;(4)满足主左理想的升链条件.     

一般环之因子幂零理想

本文称环Ω的左(右)理想A为因子幂零的,如果对于任意元素r∈Ω,均有正整数m=m(r),使得A^mr={0}.称Ω的一个左理想L为关于元素b∈Ω的左因子,如果Lb≠{0}.定理4 设R是环Ω的因子幂零右理想,那么R+ΩR是Ω的一个因子幂零理想.定理7 设Ω具有局部左因子极小条件,那么Ω的任意诣零左理想必是因子幂零左理想.本文指出因子幂零性是介于幂零性与诣零性之间的一种性质,更接近幂零性。     

右自内射环中右本质理想极大,极小条件的等价性

环论中一个熟知的结果是:当环R有单位元时,由右理想极小条件可推出右理想极大条件,但反之不然.Faith在[2]中证明在R是右自内射时,右理想极大和极小条件等价.本文中,我们研究另一类减弱的极大、极小条件:右本质理想极大和极小条件.证明了在R是右自内射的情形,它们是等价的.然后利用E.P.Armendariz的结果, 给出了QF环的一个特征,推广了Faith的相应结果. 本文中,环R均指有单位元的结合环,J记R的Jacobson根,Z_r(R)记R的右奇异(singular)理想,正则环指YOn Neumann regular,模永远指右模,若M是R-模,则     

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.     

Characterizations of Strongly Regular Rings

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TORSION THEORIES OVER N-RINGS

In this paper the author studies some properties concerned with torsion theories over an N-ring. Among other things, it is provedthat N-rings are stable.     

关于强正则环的刻画

通过单边理想是广义弱理想来刻画强正则环,证明了下列条件是等价的:①R是强正则环;②R是半素的左GP-V′-环,且每一个极大的左理想是广义弱理想;③R是半素的左GP-V′-环,且每一个极大的右理想是广义弱理想.     

Elements in one-sided unit regular rings

Let R be regular. We show that the following are equivalent:(1) R is a one sided unit regular ring. (2) For every x [euro] R, there exist an idempotente and a right or left invertible u such that x [d] eu or x [d] ue. (3) For every x [euro] R,there exists a right or left invertible u such that xu or ux is an idempotent. Moreover, we give some characterizations of one-sided unit regular rings by group inverses.     

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.     

On N0-quasi-continuous exchange rings

An associative ring R with identity is said to have stable range one if for any a,b? R with aR + bR = R, there exists y ? R such that a + by is left (equivalently, right) invertible. The main results of this note are Theorem 2: A left or right continuous ring R has stable range one if and only if R is directly finite (i.e xy = 1 implies yx = 1 for all x,y ? R), Theorem 6: A left or right N ₀o-quasi-continuous exchange ring has stable range one if and only if it is directly finite, and Theorem 12: left or right N ₀-quasi-continuous strongly π-regular rings have stable range one. Theorem 6 generalizes a well-known result of Goodearl [10], which says that a directly finite, right N _o-continuous von Neumann regular ring is unit-regular     

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.