Principally quasi-injective modules

An R-module M is called principally quasi-injective if each R-hornomorphism from a principal submodule of M to M can be extended to an endomorphism of M. Many properties of principally injective rings and quasi-injective modules are extended to these modules. As one application, we show that, for a finite-dimensional quasi-injective module M in which every maximal uniform submodule is fully invariant, there is a bijection between the set of indecomposable summands of M and the maximal left ideals of the endomorphism ring of M

Throughout this paper all rings R are associative with unity, and all modules are unital. We denote the Jacobson radical, the socle and the singular submodule of a module M by J(M), soc(M) and Z(M), respectively, and we write J(M) = J. The notation N ?^ess M means that N is an essential submodule of M.     

关于拟平坦模的几个注记

本文所有的环均指有单位元的环,模均指酉模。左R-模M称为拟内射的,如果对任意N相似文献   

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．     

纯拟内射模

本文引进了纯拟内射模的概念，讨论了该模的一些主要性质，证明了纯拟内射模保持有限直和，进一步地利用这类新模刻画了正则环的特征。     

Locally perfect commutative rings are those whose modules have maximal submodules

R denotes a commutative ring. After Bass[B], a ring R is perfect in case every module has a projective cover. A ring R is a max ring provided that every nonzero i2-module has a maximal submodule. Bass characterized perfect rings as semilocal rings with T-nilpotent Jacobson radical J, and showed they are max rings. Moreover, Bass proved that R is perfect iff R satisfies the dec on principal ideals. Using Bass' theorems, the Hamsher-Koifman ([H],[K]) characterization of max R (see (3) ?(4) below), and the characterization of max R by the author via subdirectly irreducible quasi-injective R-modules, we obtain.     

On Fully Idempotent Modules

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.     

第n根性质与根环

令E为 (MR)的自同态环 ,本文首先证明了MR 有第n根性质的充要条件是EE 有第n根性质 ,由此引进了根环的概念 ,并研究了根环的性质。特别地我们证明了如nA nB且A拟内射 ,则A B .     

强n-凝聚环

设R是一个环,n是一个正整数.右R-模M称为强n-内射的,如果从任一自由右R-模F的任一n-生成子模到M的同态都可扩张为F到M的同态;右R-模V称为强n-平坦的,如果对于任一自由右R-模F的任一n-生成子模T,自然映射VT→VF是单的;环R称为左强n-凝聚的,如果自由左R-模的n-生成子模是有限表现的;环R称为左n-半遗传的,如果R的每个n-生成左理想是投射的.本文研究了强n-内射模,强n-平坦摸及左强n-凝聚环.通过模的强n-内射性和强n-平坦性概念,作者还给出了强n-凝聚环和n-半遗传环的一些刻画.     

幺半群上的广义内射S-系(英文)

本文考虑拟内射、伪内射、核内射以及支内射S-系的性质,重点讨论这些广义内射S-系的上的线性方程组的性质.例如,A是核内射S-系当且仅当A的任意收缩核B上的任意A-相容线性方程组在B上是可解的;A是支内射S-系当且仅当同构于A的任一分支的S-系B上的任意A-相容线性方程组在B上是可解的.进而讨论了伪内射、核内射以及支内射S-系的融合余积的性质.最后给出了一个充分条件,基于此条件核内射和支内射S-系是等价的.     

On Isolated Submodules

Let R be a ring with identity and let M be a unital left R-module. A proper submodule L of M is radical if L is an intersection of prime submodules of M. Moreover, a submodule L of M is isolated if, for each proper submodule N of L, there exists a prime submodule K of M such that N ? K but L ? K. It is proved that every proper submodule of M is radical (and hence every submodule of M is isolated) if and only if N ∩ IM = IN for every submodule N of M and every (left primitive) ideal I of R. In case, R/P is an Artinian ring for every left primitive ideal P of R it is proved that a finitely generated submodule N of a nonzero left R-module M is isolated if and only if PN = N ∩ PM for every left primitive ideal P of R. If R is a commutative ring, then a finitely generated submodule N of a projective R-module M is isolated if and only if N is a direct summand of M.     

Quasi-injective dimension

Following our previous work about quasi-projective dimension [11], in this paper, we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective and Gorenstein-injective dimensions in the context of quasi-injective dimension such as the following. (a) If the quasi-injective dimension of a finitely generated module M over a local ring R is finite, then it is equal to the depth of R. (b) If there exists a finitely generated module of finite quasi-injective dimension and maximal Krull dimension, then R is Cohen-Macaulay. (c) If there exists a nonzero finitely generated module with finite projective dimension and finite quasi-injective dimension, then R is Gorenstein. (d) Over a Gorenstein local ring, the quasi-injective dimension of a finitely generated module is finite if and only if its quasi-projective dimension is finite.     

Hilbert-Schmidtness of Submodules in $H^2 (\mathbb{D}^2 )$ Containing $θ(z)−\varphi (w)$

A closed subspace $M$ of the Hardy space $H^2(\mathbb{D}^2)$ over the bidisk is called submodule if it is invariant under multiplication by coordinate functions $z$ and $w.$ Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem. This paper proves that every finitely generated submodule $M$ containing $θ(z)−\varphi(w)$ is Hilbert-Schmidt, where $θ(z),$ $\varphi(w)$ are two finite Blaschke products.     

Strongly radical supplemented modules

Z?schinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a nonlocal Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule.     

Isolated Submodules and Skew Fields

We study the generally distinct concepts of isolated submodule, honest submodule, and relatively divisible submodule for unital right R-modules, where R is an associative ring with identity. This is accomplished by studying a certain subset called the Q-torsion subset relative to a subset Q (sometimes a right ideal but not always) of R. The Q-isolator turns out to always to be a categorical closure operator and the notion of Q-honest is an `operator" but need not be a closure operator. It is shown that the notions of Q-isolated and Q-honest coincide precisely when the Q-honest operator is a closure operator and this happens precisely when all submodules are Q-honest. As a corollary, we obtain when Q = R, every submodule is honest if and only if every submodule is isolated if and only if R is a skew field. We also determine a new characterization of a right Ore domain.     

On Cofinitely Lifting and Cofinitely Weak Lifting Modules

We say that a module M is lifting if M is amply supplemented and every supplement submodule of M is a direct summand. The module M is called cofinitely lifting if it is amply cofinitely supplemented and every supplement of any cofinite submodule of M is a direct summand. In this article various properties of cofinitely lifting modules are given. In addition, a generalization of cofinitely lifting modules is investigated.     

Infinite Direct Sums of Lifting Modules

A module M over a ring R is called a lifting module if every submodule A of M contains a direct summand K of M such that A/K is a small submodule of M/K (e.g., local modules are lifting). It is known that a (finite) direct sum of lifting modules need not be lifting. We prove that R is right Noetherian and indecomposable injective right R-modules are hollow if and only if every injective right R-module is a direct sum of lifting modules. We also discuss the case when an infinite direct sum of finitely generated modules containing its radical as a small submodule is lifting.     

FP-内射环的一个特征

本文首次利用投射模给出了右ＦＰ-内射环的一个外部特征，即Ｒ为右ＦＰ-内射环当且仅当投射左Ｒ-模的有限生成子模为闭子模。     

广义拟内射模的自同态环

本文的目的,是推广[1]中定理1.22和[2]中命题1(1).我们得到:设R是环,且Q=End_R(M),其中M是广义拟内射模.那么有(1)J(Q)=Z(Q);(2)Q/J(Q)是Von Neumann正则环.     

Nouvelles caracterisations des v-anneaux et des anneaux pseudo-frobeniusiens 1

A class of A-modules is socle-fine if for any A-modules M and N in this class, M and N are isomorphic if and only if socle of M and socle of N are isomorphic. A ring A is a left V-ring if and only if the class of indecomposable quasi-injective A-modules with large socle is socle-fine. A ring A is a left V-ring noetherian if and only if the class of quasi-injective /4-modules with large socle is socle-fine. A ring A is Pseudo-Frobenius if and only if A is a left, cogenerator and the class of projective A-modules is socle-fine.     

MODULES FOR WHICH EVERY SUBMODULE HAS A UNIQUE COCLOSURE

ABSTRACT

P. F. Smith studied modules in which every submodule has a unique closure and called them UC modules. In this paper we consider modules with the dual property viz., those in which every submodule has a unique coclosure and call such modules UCC modules. Unlike closures, a coclosure of a submodule of a module may not always exist and even if it exists, it may not be unique. We investigate the conditions under which a module is a UCC module. We prove that UCC modules are closed under factor modules and coclosed submodules. We also investigate their properties and their relation to non-cosingular modules, copolyform modules, and codimension modules. We end this paper with the dual of Smith's result on dimension modules.