投射根和亚投射模

本文研究了投射根和亚投射模的性质.利用Noether环、Noether模的性质给出了亚投射模、投射根是投射的条件,环的投射根为零、幂等的条件.最后,给出了一些例子.本文的大部分结果是陈焕艮等人的一系列主要结果的推广.     

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.     

具有素中心环的若干性质(英文)

给出了素中心环的若干新的性质 ,在具有素中心的条件下 ,我们证明了 :环的幂零元与强幂零元是一致的 ;环的素根与诣零根是相同的 ;环的满自同态是自同构 ;对于每个 a∈ R,序列 Ra Ra2 …是稳定的当且仅当对于每个 a∈ R,存在自然数 n使得 an是一个正则元 .研究了某些具有素中心的特殊环     

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.     

Γ-环的Z-正则根

<正> 本文对Γ-环A定义了一种Z-正则性,证明了任-Γ-环A有一个最大的Z-正则理想Z(A);Z(A)是Kypow意义下的根,称之为Z-正则根;此根对理想还是可遗传的.这个结果是结合环,弱Γ_N-环和Γ-环中关于Neumann正则性和正则根的推广.     

环的弱遗传类和本质扩张

§1.符号及引理所有的环均指结合环。所谓根类或半单类,是Kurosh及Armitsur意义下的相应概念。遗传类、正则类、同态闭类、(弱)特殊类及遗传根、特殊根、超幂零根等概念参阅[6]与[7]。     

分次Abel正则环的结构

首先给出了 gr-正则环为分次除环的两个充要条件 ,其次讨论了分次正则环r G(R)和分次 Jacobson根 JG(R)之间的关系 ,最后给出了分次 Abel正则环的结构定理 .     

拟投射生成元自同态环的理想格

设Δ是忠实拟投射生成元 R~U 的自同态环,本文证明了Δ的理想格与 R 的迹理想 J 中的零化理想格之间的同构,讨论了 R 的本原性与Δ的本原性,J 的根与Δ的根之间的相互关系.     

几乎半正则环（英文）

设R是环,J(R)是R的Jacobson根.R的元素a称为半正则元,如果存在正则元b∈R使得a-b∈J(R).环R称为几乎半正则环,如果对R的任意元a,有a或者1-a是半正则的.本文引入了几乎半正则环作为VNL-环和半正则环的推广.构造了一些例子,证明了几乎半正则环是置换环;将半正则环的许多性质推广到了几乎半正则环上.     

有幺单Γ－环类所确定的上根

结合环根论中由具有单位元的单环类所确定的上根就是Ｂｒｏｗｎ－ＭｃＣｏｙ根．本文指出在Γ－环中，鉴于Γ－环单位元的复杂性质使得有幺单Γ－环类确定的上根演变成８个，进而研究了这些根之间及这些根与Γ－环其它根之间的大小关系．     

Invariant subrings of separable algebras

This paper gives a necessary and sufficient condition that the ring of invariants of every group of automorphisms of every projective, separable, commutative algebra over a given commutative ring is itself a union of separable, projective subalgebras. Rings satisfying the condition include products of connected rings, von Neumann regular rings, and some rings of functions.     

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R) is a commutative regular ring with J(R) nil,where J(R) is the Jacobson radical of R.     

Radicals of matrix rings generated by a companion matrix

The radicals of full matrix rings as well as structural matrix rings have been studied extensively and there is a well-developed theory for both. Here we initiate the radical theory for another class of matrix rings over a given ring; the matrix ring generated by the companion matrix of a polynomial over the ring.     

On the Left Perpendicular Category of the Modules of Finite Projective Dimension

In this article, we characterize several properties of commutative noetherian local rings in terms of the left perpendicular category of the category of finitely generated modules of finite projective dimension. As an application, we prove that a local ring is regular if (and only if) there exists a strong test module for projectivity having finite projective dimension. We also obtain corresponding results with respect to a semidualizing module.     

Analogue of G. Higman's Theorem for Commutative Torsion-Free Rings

In this paper we consider finite rank torsion-free rings, which have almost regular automorphisms of prime order (a non-trivial automorphism is called almost regular if it has only trivial fixed points, i.e. zero and the elements of a ring linear dependent on its identity). The main result of this paper is the analogue of G. Higman's known Theorem [1] on almost regular automorphism for commutative finite rank torsion-free rings.     

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.     

Tensor powers of symmetric algebras

We prove that over an integral domain a module is projective iff an appropriate tensor power of its symmetric algebra is an integral domain. Further, we show that contracting parts of primary decom­positions of the zero ideal in appropriate tensor powers of a symmetric algebra one obtains families of ideals canonically associated to a mod­ule, having the same radical as Fitting ideals. More precisely, we prove that those new ideals lie between the annihilators of exterior powers of the module and their radicals. An immediate consequence of our re­sults is a way to recover the radicals of Fitting ideals of a module from the symmetric algebra of that module (with its grading forgotten).     

Radicals of Jordan rings connected with alternative rings

Subject to a certain restriction on the additive group of an alternative ring A, we prove that R(A)=R(A⁽⁺⁾), where A⁽⁺⁾ is a Jordan ring and R is one of the following radicals: the Jacobson radical, the upper nil-radical, the locally nilpotent radical, or the lower nil-radical. For the proof of these relationships Herstein's well-known construction for associative rings is generalized to alternative rings.