Semi-Endosimple Modules and Some Applications

Asian-European Journal of Mathematics, 2017

This paper provides the several homological characterization of perfect rings and semi-simple rings in terms of semi-projective modules. We investigate whether Hopkins–Levitzki Theorem extend to semi-projective module i.e. whether there exists an artinian semi-projective module which are noetherian. Unfortunately, the answer we have is negative; counter example is given. However, it is shown that the answer is positive for certain large classes of semi-projective modules in Proposition 2.26. We have discussed the summand intersection property, summand sum property for semi-projective modules. Apart from this, we have introduced the idea of [Formula: see text]-hollow modules, also several necessary and sufficient conditions are established when the endomorphism rings of a semi-projective modules is a local ring.

Journal of Pure and Applied Algebra, 2004

A module M is said to satisfy the condition (˝ *) if M is a direct sum of a projective module and a quasi-continuous module. By Huynh and Rizvi (J. Algebra 223 (2000) 133; Characterizing rings by a direct decomposition property of their modules, preprint 2002) rings over which every countably generated right module satisÿes (˝ *) are exactly those rings over which every right module is a direct sum of a projective module and a quasi-injective module. These rings are called right˝ *-semisimple rings. Right˝ *-semisimple rings are right artinian. However, in general, a right˝ *-semisimple rings need not be left˝ *-semisimple. In this note, we will prove a ring-direct decomposition theorem for right and left˝ *-semisimple rings. Moreover, we will describe the structure of each direct summand in the obtained decomposition of these rings.

Publicacions Matemàtiques, 2013

We carry out a study of rings R for which Hom R (M, N) = 0 for all nonzero N ≤ M R. Such rings are called retractable. For a retractable ring, Artinian condition and having Krull dimension are equivalent. Furthermore, a right Artinian ring in which prime ideals commute is precisely a right Noetherian retractable ring. Retractable rings are characterized in several ways. They form a class of rings that properly lies between the class of pseudo-Frobenius rings, and the class of max divisible rings for which the converse of Schur's lemma holds. For several types of rings, including commutative rings, retractability is equivalent to semi-Artinian condition. We show that a Köthe ring R is an Artinian principal ideal ring if and only if it is a certain retractable ring, and determine when R is retractable.

Journal of pure and applied algebra, 1992

Proceedings of the American Mathematical Society, 1995

We use the concept of dual Goldie dimension and a characterization of semi-local rings due to to find some classes of modules with semi-local endomorphism ring. We deduce that linearly compact modules have semi-local endomorphism ring, cancel from direct sums and satisfy the n th root uniqueness property. We also deduce that modules over commutative rings satisfying AB5* also cancel from direct sums and satisfy the n th root uniqueness property.

Pacific Journal of Mathematics

arXiv: Rings and Algebras, 2020

Recently, in a series of papers "simple" versions of direct-injective and directprojective modules have been investigated (see, [4], [12], [13]). These modules are termed as "simple-direct-injective" and "simple-direct-projective", respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right Hrings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simpledirect-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given.

Algebra Colloquium, 2008

Let U be a submodule of a module M. We call U a strongly lifting submodule of M if whenever M/U=(A+U)/U ⊕ (B+U)/U, then M=P ⊕ Q such that P ≤ A, (A+U)/U=(P+U)/U and (B+U)/U=(Q+U)/U. This definition is a generalization of strongly lifting ideals defined by Nicholson and Zhou. In this paper, we investigate some properties of strongly lifting submodules and characterize U-semiregular and U-semiperfect modules by using strongly lifting submodules. Results are applied to characterize rings R satisfying that every (projective) left R-module M is τ (M)-semiperfect for some preradicals τ such as Rad , Z2 and δ.

Sbornik: Mathematics, 1998

Let A be a hereditary Noetherian prime ring that is not right primitive. A complete description of π-injective A-modules is obtained. Conditions under which the classical ring of quotients of A is a π-projective A-module are determined. A criterion for a right hereditary right Noetherian prime ring to be serial is obtained. Bibliography: 8 titles. All rings in this paper are assumed to be associative and with a non-zero identity element. Expressions such as 'a Noetherian ring' mean that the corresponding right-hand and left-hand conditions hold. The Jacobson radical of a module M is denoted by J(M). A module is hereditary if all its submodules are projective. A module is uniserial if any two of its submodules are comparable with respect to inclusion. A module M is serial if M is a direct sum of uniserial modules. A uniserial Noetherian domain A is complete if the ring A is complete with respect to the J(A)-adic topology. A module M is projective with respect to a module N (or N-projective) if for each epimorphism h: N → N and each homomorphism f : M → N there exists a homomorphism f : M → N such that hf = f. A module that is projective with respect to itself is called a quasiprojective (or self-projective) module. A module M is π-projective (see [1], p. 359) if for arbitrary submodules U and V of M such that U + V = M there exists an endomorphism f of M such that f(M) ⊆ U and (1 − f)(M) ⊆ V. If all factor modules of a module M are indecomposable, then M is clearly π-projective. Hence all uniserial modules are π-projective. Each quasiprojective module is π-projective [1], p. 359. Each quasicyclic Abelian group C(p ∞) is a π-projective non-quasiprojective module over the ring of integers. Recall that each Noetherian prime ring A has a simple Artinian classical ring of quotients Q, and Q A is an injective non-singular right (left) A-module. Conditions ensuring that the classical ring of quotients of a hereditary Noetherian prime, but not right primitive ring A is a quasiprojective right A-module are described in [2], Theorem 8. In this connection we present the following Theorems 1 and 2, which are the first two main results of this paper.

Carpathian Mathematical Publications, 2020

As a proper generalization of injective modules in term of supplements, we say that a module $M$ has the property (ME) if, whenever $M\subseteq N$, $M$ has a supplement $K$ in $N$, where $K$ has a mutual supplement in $N$. In this study, we obtain that $(1)$ a semisimple $R$-module $M$ has the property (E) if and only if $M$ has the property (ME); $(2)$ a semisimple left $R$-module $M$ over a commutative Noetherian ring $R$ has the property (ME) if and only if $M$ is algebraically compact if and only if almost all isotopic components of $M$ are zero; $(3)$ a module $M$ over a von Neumann regular ring has the property (ME) if and only if it is injective; $(4)$ a principal ideal domain $R$ is left perfect if every free left $R$-module has the property (ME)