关于主伪内射模

对于环R．一个右R模被叫做主伪内射模。若每一个从M的主子模到M的单同态可以扩张为M的自同态．主伪内射是主拟内射的推广．在本文中,我们给出了一些主伪内射的性质并讨论什么情况下主伪内射模是主拟内射模的问题．     

Injective and automorphism-invariant non-singular modules

Every automorphism-invariant non-singular right A-module is injective if and only if the factor ring of the ring A with respect to its right Goldie radical is a right strongly semiprime ring.     

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。     

有链条件的非奇异环

本文先引入 C:一环、Br ·环等等若干环类并给出它们成为 G ol ide环的一个充要 条件.本文还给出半质右G old ie 环为左G ol id环的一个刻划     

Applications of Duality to the Pure-injective Envelope

Given an R-T-bimodule _R K _T and R-S-bimodule _R M _S , we study how properties of _R K _T affect the K-double dual M** = Hom _T [Hom _R (M, K), K] considered as a right S-module. If _R K is a cogenerator, then for every R-S-bimodule, the natural morphism Φ _M : M → M** is a pure-monomorphism of right S-modules. If _R K is the minimal (injective) cogenerator and K _T is quasi-injective, then M ^** is a pure-injective right S-module. If _R K is the minimal (injective) cogenerator, and T = End _R K it is shown that K _T is quasi-injective if and only if the K-topology on R is linearly compact. If the _R K-topology on R is of finite type, then the natural morphism Φ _R : R → R** is the pure-injective envelope of R _R as a right module over itself. The author is partially supported by NSF Grant DMS-02-00698.     

Rings all of whose right ideals are U-modules

The notion of a U-module was introduced and thoroughly investigated in [11 Ibrahim, Y., Yousif, M. F. (2017). U-modules. Comm. Algebra, doi:https://doi.org/10.1080/00927872.2017.1339064.[Crossref] , [Google Scholar]] as a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant modules. In this paper a right R-module M is called a U*-module if every submodule of M is a U-module, and a ring R is called a right U*-ring if R_R is a U*-module. We show that M is a U*-module iff whenever A and B are submodules of M with A?B and A∩B = 0, A⊕B is a semisimple summand of M; equivalently M = X⊕Y, where X is semisimple, Y is square-free, and X &; Y are orthogonal. In particular, a ring R is a right U*-ring iff R is a direct product of a square-full semisimple artinian ring and a right square-free ring. Moreover, right U*-rings are shown to be directly-finite, and if the ring is also an exchange ring then it satisfies the substitution property, has stable-range 1, and hence is stably-finite. These results are non-trivial extensions of similar ones on rings all of whose right ideals are either quasi-continuous or auto-invariant.     

A companion ideal of a multiplication module

Let M _R be a faithful multiplication module, where R is a commutative ring. As defined by Anderson, this ideal has proved to be useful in studying multiplication modules. First of all a cancellation law involving M and the ideals contained in is proved. Among various applications given, the following result is proved:: There exists a canonical isomorphism from onto such that for any ( Hom _R(M,M), x ( M, a ( (M), (xa) = x.(()(a). As an application of this later result it is proved that M is quasi-injective if and only if (M) is quasi-injective.