Semisimple Rings Research Articles

Relative injective modules, superstability and noetherian categories

We study classes of modules closed under direct sums, [Formula: see text]-submodules and [Formula: see text]-epimorphic images where [Formula: see text] is either the class of embeddings, RD-embeddings or pure embeddings. We show that the [Formula: see text]-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and [Formula: see text]-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah’s non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability.

Remarks on rings with u − 1 quasi-nilpotent for each unit u

This paper is concerned with the investigation of those rings [Formula: see text] whose invertible elements are sums [Formula: see text] for an element [Formula: see text] such that [Formula: see text] is invertible for all invertible [Formula: see text] with [Formula: see text]. UJ-rings and UQ-rings (and hence UU-rings) are examples of such rings. We prove that such rings are Dedekind finite, and for semisimple rings [Formula: see text], [Formula: see text] In case [Formula: see text] is also a semipotent ring, it is shown that [Formula: see text] is a Boolean ring.

On modules in which every proper finitely generated submodule is a subset of a kernel of a nonzero endomorphism

We study an R-module M such that for every proper finitely generated submodule K of M, Hom R ( M / K , M ) ≠ 0 . Such modules are called co-finite-retractable (simply CFR). We consider when factor modules, direct sums and direct summands of CFR modules are CFR. Also, we investigate the injective hull of CFR modules. We explore certain properties of a ring R such that every (cyclic, free) module is CFR. Also, we show that R is a semisimple ring if and only if every R-module is regular if and only if every CFR module over R is regular.

Restricted-finite groups with some applications in group rings

We carry out a study of groups $G$ in which the index of any infinite subgroup is finite. We call them restricted-finite groups and characterize finitely generated not torsion restricted-finite groups. We show that every infinite restricted-finite abelian group is isomorphic to $ \mathbb{Z}\times K$ or $\mathbb{Z}_{p^\infty}\times K$, where $K$ is a finite group and $p$ is a prime number. We also prove that a group $G$ is infinitely generated restricted-finite if and only if $G = AT$ where $A$ and $T$ are subgroups of $G$ such that $A$ is normal quasi-cyclic and $T$ is finite. As an application of our results, we show that if $G$ is not torsion with finite $G'$ and the group-ring $RG$ has restricted minimum condition, then $R$ is a semisimple ring and $G\cong T\rtimes\mathbb{Z} $, where $T$ is finite whose order is unit in $R$. The converse is also true with certain conditions including $G = T\times \mathbb{Z} $.

When every S-flat module is (flat) projective

Let R be a commutative ring with identity and S a multiplicative subset of R. The aim of this paper is to study the class of commutative rings in which every S-flat module is flat (resp., projective). An R-module M is said to be S-flat if the localization of M at S, MS , is a flat RS -module. Commutative rings R for which all S-flat R-modules are flat are characterized by the fact that R/Rs is a von Neumann regular ring for every s ∈ S . While, commutative rings R for which all S-flat R-modules are projective are characterized by the following two conditions: R is perfect and the Jacobson radical J(R) of R is S-divisible. Rings satisfying these conditions are called S-perfect. Moreover, we give some examples to distinguish perfect rings, S-perfect rings, and semisimple rings. We also investigate the transfer results of the “S-perfectness” for various ring constructions, which allows the construction of more interesting examples.

Pure C3 modules

Within the paper, we study the class of modules in which the direct sum of two disjoint summands that is a pure submodule is also a summand. This class of modules is considered [Formula: see text] modules, a generalization of [Formula: see text] modules. In addition to pure-injective and [Formula: see text] modules that belong to the class of [Formula: see text] modules, which also include the semisimple, continuous, indecomposable, and regular modules. We gave new characterizations of many rings in respect of [Formula: see text] modules, namely semisimple rings, pure-semisimple rings, von Neumann regular rings, Noetherian rings, pure-hereditary rings, pure-[Formula: see text]-rings, etc. Moreover, we also discuss the [Formula: see text] envelope and [Formula: see text] cover of a module and introduce the pure continuous modules as the generalization of the continuous modules and also introduce the pure quasi-continuous modules as the generalization of the quasi-continuous modules.

Structure of semisimple rings in reverse and computable mathematics

Structure of semisimple rings in reverse and computable mathematics

Rings whose modules satisfy chain conditions on essential endosubmodules

We study rings R such that all (or certain classes) of right R-modules satisfy chain conditions on essential endosubmodules. It is proved that if all right R-modules have DCC on essential endosubmodules, then R is a right pure semisimple ring, and if all right R-modules have both ACC and DCC on essential endosubmodules, then R is of finite representation type. Rings R such that all right R-modules have ACC on essential endosubmodules are discussed, and it is shown, in particular, that if such a ring R is right Artinian, then R is left pure semisimple.

On distribution of the number of semisimple rings of order at most x in an arithmetic progression

Let \ell and q denote relatively prime positive integers. In this article, we derive the asymptotic formula for the summation \begin{align*} \sum_{n\leq x\atop n\equiv \ell \pmod q}S(n), \end{align*} where S(n) denotes the number of non-isomorphic finite semisimple rings with n elements.

Graded (quasi-)Frobenius rings

In order to study graded Frobenius algebras from a ring theoretical perspective, we introduce graded quasi-Frobenius rings, graded Frobenius rings and a shift-version of the latter ones, and we investigate the structure and representations of such objects. We need to revisit graded simple graded left Artinian rings, graded semisimple rings, and to provide graded versions of certain results concerning the Jacobson radical, the singular radical, and their connection to finiteness conditions and injectivity. We prove a structure result for (shift-)graded Frobenius rings.

On simple submodules and C-rings

In this paper; we will study the relationship between simple submodules of M and the C-ring. A ring R is called C-ring if every torsion module M over R has simple submodules. We investigated that if M is a torsion module over a semisimple ring; so, R is C-ring. Also, we study C-ring when R is Noetherian and T(M) = M such that any direct sum of modules with (SIP) has (SSP) implies that R is C- ring. We proved that if M is a torsion module and M = W + W′, this means R is C-ring where W and W′ are direct sum of M.

Primitive idempotents in a semisimple ring

Let [Formula: see text] be distinct primes, [Formula: see text] odd. Let [Formula: see text], where [Formula: see text] be an integer. In this paper, it is observed that the explicit expressions of primitive idempotents from [Formula: see text] are sufficient to compute the explicit expressions of primitive idempotent in semisimple ring [Formula: see text]. It is also shown that the results obtained in [A. Sahni and P. T. Sehgal, Minimal cyclic codes of length [Formula: see text], Finite Fields Appl. 18(5) (2012) 1017–1036; P. Kumar and S. K. Arora, [Formula: see text]-mapping and primitive idempotents in semisimple ring [Formula: see text], Comm. Algebra 41(10) (2013) 3679–3694] are simple corollaries to the results obtained in the paper.

Restricted minimum condition for group-rings and matrix extensions

A ring R is said to have right restricted minimum condition ( for short) if R/A is an Artinian right R-module for any essential right ideal A of R. We study the for matrix extensions of a ring R and for the group-ring RG. Among other things, it is proven that having r.RMC is a Morita invariant property for rings. For the upper triangular n by n matrix ring over R has r.RMC if and only if RR is Artinian. If G is a not torsion abelian group then RG has r.RMC if and only if R is a semisimple ring and where H is a finite group whose order is invertible in R. If X is a completely regular topological space, then the ring C(X) has if and only if X is finite. Many examples of rings with r.RMC are presented.

Modules and semi-simple rings. I.

Modules and semi-simple rings. I.

The general theory of linear equation systems over semi-simple rings

The general theory of linear equation systems over semi-simple rings

δ-dual codes over finite commutative semi-simple rings

δ-dual codes over finite commutative semi-simple rings

THE PROBABILITY OF ZERO MULTIPLICATION IN FINITE RINGS

AbstractLet R be a finite ring and let ${\mathrm {zp}}(R)$ denote the nullity degree of R, that is, the probability that the multiplication of two randomly chosen elements of R is zero. We establish the nullity degree of a semisimple ring and find upper and lower bounds for the nullity degree in the general case.

Reverse mathematics and semisimple rings

Reverse mathematics and semisimple rings

The soft Jacobson radical of a commutative ring

In this paper, the notion of the soft Jacobson radical of a ring is defined. A relationship between the soft Jacobson radical of a ring and Jacobson semisimple ring is established. Some properties of this notion have been studied under homomorphism.

On automorphism group of orthogonality graph of finite semisimple rings

The orthogonality graph of a ring R, written as O(R), is an undirected simple graph with vertex set the set of all nonzero two-sides zero-divisors of R, and for distinct there exists an edge joining x and y if and only if In this paper, we describe the automorphism group of O(R) when R is a finite semisimple ring.