Minus Partial Order in Regular Modules

2018

This work is devoted to the investigation of module decompositions which arise from Rickart modules, socle and radical of modules. In this regard, the structure and several illustrative examples of inverse split modules relative to the socle and radical are given. It is shown that a module \(M\) has decompositions \(M=\mathrm{Soc}(M) \oplus N\) and \(M=\mathrm{Rad}(M) \oplus K\) where \(N\) and \(K\) are Rickart if and only if \(M\) is \(\mathrm{Soc}(M)\)-inverse split and \(\mathrm{Rad}(M)\)-inverse split, respectively. Right \(\mathrm{Soc}(\cdot)\)-inverse split left perfect rings and semiprimitive right hereditary rings are determined exactly. Also, some characterizations for a ring \(R\) which has a decomposition \(R=\mathrm{Soc}(R_R)\oplus I\) with \(I\) hereditary Rickart module are obtained.

2009

In this paper, the explicit form of maximal elements, known as shorted operators, in a subring of a von Neumann regular ring has been obtained. As an application of the main theorem, the unique shorted operator (of electrical circuits) which was introduced by Anderson-Trapp has been derived.

2018

Let \(R\) be a ring. A right \(R\)-module \(M\) is called \(\mathrm{d}\)- Rickart if for every endomorphism \(\varphi\) of \(M\), \(\varphi(M)\) is a direct summand of \(M\) and it is called \(\mathrm{wd}\)- Rickart if for every nonzero endomorphism \(\varphi\) of \(M\), \(\varphi(M)\) contains a nonzero direct summand of \(M\). We begin with some basic properties of \(\mathrm{(w)d}\)-Rickart modules. Then we study direct sums of \(\mathrm{(w)d}\)-Rickart modules and the class of rings for which every finitely generated module is \(\mathrm{(w)d}\)-Rickart. We conclude by some structure results.

Bulletin of the Australian Mathematical Society

It is shown that a von Neumann regular ring R is left seif-injective if and only if every finitely generated torsion-free left R-module is projective. It is further shown that a countable self-injective strongly regular ring is Artin semi-simple.

Communications in Algebra, 2007

Journal of Algebra and Its Applications, 2018

Let [Formula: see text] be a ring. Motivated by a generalization of a well-known minus partial order to Rickart rings, we introduce a new relation on the power set [Formula: see text] of [Formula: see text] and show that this relation, which we call “the minus order on [Formula: see text]”, is a partial order when [Formula: see text] is a Baer ring. We similarly introduce and study properties of the star, the left-star, and the right-star partial orders on the power sets of Baer ∗-rings. We show that some ideals generated by projections of a von Neumann regular and Baer ∗-ring [Formula: see text] form a lattice with respect to the star partial order on [Formula: see text]. As a particular case, we present characterizations of these orders on the power set of [Formula: see text], the algebra of all bounded linear operators on a Hilbert space [Formula: see text].

Let R be an arbitrary ring with identity and M a right R-module with S = EndR(M). In this paper we introduce dual π-Rickart modules as a generalization of π-regular rings as well as that of dual Rickart modules. The module M is said to be dual π-Rickart if for any f ∈ S, there exist e 2 = e ∈ S and a positive integer n such that Im f n = eM. We prove that some results of dual Rickart modules can be extended to dual π-Rickart modules for this general settings. We investigate relations between a dual π-Rickart module and its endomorphism ring.

Proceedings of the Japan Academy, Series A, Mathematical Sciences, 1987

Introduction. Let R be a fixed (not necessarily commutative) ring. Throughout this nte, we are concerned with left R-modules M, A, H, Like in'Goldie [1], we shall use the following terminology. A non-zero submodule K of M is called essential in M (or M is an essential extension

Colloquium Mathematicum, 2015

A module M over a ring R satises the restricted minimum condition (RMC) if M=N is Artinian for every essential submodule N of M. A ring R satises right RMC if R R satises RMC as a right module. It is proved that (1) a right semiartinian ring R satises right RMC if and only if R=Soc(R) is Artinian, (2) if a semilocal ring R satises right RMC and Soc(R) = 0, then R is Noetherian if and only if the socle length of E(R=J(R)) is at most !, and (3) a commutative ring R satises RMC if and only if R=Soc(R) is Noetherian and every singular module is semiartinian.