关于基本可数生成子模的扩张性

称左R-模M是ecg-扩张模,如果M的任意基本可数生成子模是M的直和因子的基本子模.在研究了ecg-扩张模的基本性质的基础上,本文证明了对于非奇异环R,所有左R-模是ecg-扩张模当且仅当所有左R-模是扩张模.同时我们还用ecg-拟连续模刻画了Noether环和Artin半单环.     

On semilocal rings

We give several characterizations of semilocal rings and deduce that rationally closed subrings of semisimple artinian rings are semilocal, that artinian modules have semilocal endomorphism rings, and that artinian modules cancel from direct sums. Dedicated to the memory of Pere Menal     

局部Noether模的相对连续性特征

设M是左R-模,本文证明了M是局部Noether的当且仅当σ[M]中的任意M-内射左R-模的直和是S∧2-连续的（S∧2-拟连续的）。     

On the endomorphism ring of a module noetherian with respect to a torsion theory

IfM is a module torsionfree and noetherian with respect to a torsion theory, ifS is the endomorphism ring ofM, and ifL(S) is the ideal ofS consisting of all endomorphisms with large kernels, thenL(S) is nilpotent and a bound on the index of nilpotency ofL(S) is given.     

The rational loewy series and nilpotent ideals of endomorphism rings

Sufficient conditions are given, in module-theoretic terms, for the idealN(S) of the endomorphism ringS of a moduleM consisting of the endomorphisms with essential kernel to be nilpotent. This extends in a natural way several known results on the nilpotency ofN(S). WhenM is a quasi-injective module such thatS is right noetherian, it is shown thatS is right artinian if and only ifM has a finite rational Loewy series whose length is, in this case, equal to the index of nilpotency ofN(S). The author has been partially supported by the CAICYT.     

A generalization of right pci rings

By a well-known result of Osofsky [6, Theorem] a ring R is semisimple (i.e. R is right artinian and the Jacobson radical of R is zero) if and only if every cyclic right R-module is injective. Starting from this, a larger class of rings has been introduced and investigated, namely the class of right PCI rings. A ring R is called right PCI if every proper cyclic right R- module is injective (proper here means not being isomorphic to RR). By [l] and [Z], a right PCI ring is either semisimple or it is a right noetherian, right hereditary simple ring. The latter ring is usually called a right PCI domain. In this paper we consider the similar question in studying rings whose cyclic right modules satisfy some decomposition property. The starting point is a theorem recently proved in 13, Theorem 1.1): A ring R is right artinian if and only if every cyclic right R- module is a direct sum of an injective module and a finitely cogenerated module.     

ON DISTRIBUTIVE MODULES AND LOCALLY DISTRIBUTIVE RINGS

Distributive modules over artinian rings are characterized via module diagrams,andit is shown that a left artinian ring is(two-sided)locally distributive in case its leftindecomposable injective modules and projective modules are distributive.This latter resultis used to show that a locally distributive artinian ring and the endomorphism ring of itsminimal cogenerator have identical diagrams.     

Simple artinian rings as sums of nilpotent subrings

Decompositions of simple artinian rings as additive sums of nilpotent subrings are considered. In particular, necessary and sufficent conditions for minimal decompositions are found in terms of the underlying division ring. This is used to prove that any algebra with 1 can be unitarily embedded into a simple algebra with 1 which is a sum of four subalgebras with zero multiplication and also into a simple algebra which is a sum of three nilpotent subalgebras of degree 3. Since our proof is constructive and is based on simple artinian rings, the latter result can be viewed as an extension and a strengthened version of Bokut's theorem, [2].     

On completions of non-commutative noetherian rings

It is shown that if I is an ideal of a ring R ,and I has a centralising set of generators then the I-adic completion [Rcirc] is left noetherian if either R/I is left artinian or R is left noetherian.     

Modules semicocritical with respect to a torsion theory and their applications

The concepts of primitive ideal and semicocritical module with respect to a torsion theory are studied and related to the structure of torsionfree injective modules. Applications are made to the study of (1) composition series with respect to a torsion theory and (2) the structure of endomorphism rings of torsionfree modules. These results are natural generalizations of the properties of certain modules over (noetherian) rings with Krull dimension.     

On Finite Basis Property for Joins of Varieties of Associative Rings

If all finitely generated rings in a variety of associative rings satisfy the ascending chain condition on two-sided ideals, the variety is called locally weak noetherian. If there is an upper bound on nilpotency indices of nilpotent rings in a variety, the variety is called a finite index variety. We prove that the join of a finitely based locally weak noetherian variety and a variety of finite index is also finitely based and locally weak noetherian. One consequence of this result is that if an associative ring variety is connected by a finite path in the lattice of all associative ring varieties to a finitely based locally weak noetherian variety then such variety is also finitely based and locally weak noetherian.     

广义逆多项式模的co-Hopf性质

设R是结合环(可以没有单位元),(S,≤)是严格全序幺半群,序≤是Artin的且对任意s∈S,有0≤s,则对任意具有性质(F)的左R-模M,[MS,≤]是co-Hopf左[[RS,≤]]一模当且仅当M是co-Hopf左R-模.     

Nil and strongly nil derivations

In a semiprime algebra R over a field of characteristic p ≥ 0, a derivation d is called nil if for each x ? R there is n such that dⁿx = 0. It is called strongly nil if it is induced by a nil element in the symmetric ring of quotients. The first result of this paper is an intrinsic criterion on a nil derivation being strongly nil. We then use this criterion to establish some relation between a nilpotent derivation and a strongly nilpotent derivation after a preliminary discussion of the nilpotency, or index of nilpotency, of a nilpotent derivation. Examples are given later in the paper to show that in some sense these results are best possible. This paper is self contained.     

On the existence of covers by injective modules relative to a torsion theory

Let R be a ring with identity. In this note we study covers of left R-modules by r-injectives left R-modules, where r is a hereditary torsion theory defined in the category of all left R-modules and all R-morphisms. When R is an artinian commutative ring, a complete answer about the existence of such covers for every R-module is given. In case that T is a centrally splitting torsion theory, we can characterize those T for which every left R-module has a T-injective cover. Also we analyze R-modules such that the injective and the T-injective cover are the same. At the end of this note we relate the concepts of colocalization and cover     

On nil subrings

The nil subrings of rings which satisfy certain ascending chain condition on anihilators are shown to be nilpotent. Professor J. Levitzki died in 1956. This result has been found among his papers and was arranged for publication by S. A. Amitsur.     

INJECTIVE PRECOVERS AND MODULES OF GENERALIZED INVERSE POLYNOMIALS

Let (S,≤) be an ordered set. Recall that (S,≤) is artinian if every strictly decreasingsequence of elements of S is ?nite, and that (S,≤) is narrow if every subset of pairwiseorder-incomparable elements of S is ?nite. Let S be a commutative monoid. Unl…     

A NOTE ON LEFT STROUGLY NIL AND LEFT STROUGLY NILPOTENT

Abstract

In [2] van der Walt called a left ideal L of a ring A, left strongly nil, if given 1 ε L and k ε K, K a left ideal. there is an n such that (1+k)ⁿ ε K. L is called left strongly nilpotent if for any left ideal K there exists an m such that (L+K)^m ? K. In this paper we will prove that if A is a left artinian ring (not necessarily with unity) then every left strongly nil left ideal is left strongly nilpotent. This result is a generalization of the main theorem of [2].     

Characterizations of some rings with <Emphasis Type="Italic">C</Emphasis>-projective, <Emphasis Type="Italic">C</Emphasis>-(FP)-injective and <Emphasis Type="Italic">C</Emphasis>-flat modules

Let R be a commutative ring and C a semidualizing R-module. We investigate the relations between C-flat modules and C-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.     

Abelian Groups with Annihilator Ideals of Endomorphism Rings

We describe the periodic groups whose endomorphism rings satisfy the annihilator condition for the principal left ideals generated by nilpotent elements. We prove that torsion-free reduced separable, vector, and algebraically compact groups have endomorphism rings with the annihilator condition for the principal left (right) ideals generated by nilpotent elements if and only if these rings are commutative. We show that the almost injective groups (in the sense of Harada) are injective, i.e. divisible.     

The prüfer rings that are endomorphism rings of artinian modules

In this paper we characterize the (commutative) Priifer rings that can be realized as endomorphism rings of artinian modules over arbitrary associative rings with identity (Theorem 4.7). This characterization is obtained by determining the structure of ∑-pure-injective modules over Prufer rings (Theorems 3.4 and 3.5)