Class Of Rings Research Articles

Compressed intersection annihilator graph

Let R be a commutative ring with a non-zero identity. In this paper, we define a new graph, the compressed intersection annihilator graph, denoted by $IA(R)$, and investigate some of its theoretical properties and its relation with the structure of the ring. It is a generalization of the torsion graph $\Gamma_{R}(R)$. We study classes of rings for which the equivalence between the set of zero-divisors of $R$ being an ideal and the completeness of $IA(R)$ holds. We also study the relation between $\Gamma_{R}(R)$ and $IA(R)$. In addition, we show that if the compressed intersection annihilator graph of a ring $R$ is finite, then there exists a subring $S$ of $R$ such that $IA(S)\cong IA(R)$. Also, we show that the compressed intersection annihilator graph will never be a complete bipartite graph. Besides, we show that the graph $IA(R)$ with at least three vertices is connected and its diameter is less than or equal to three. Finally, we determine the properties of the graph in the cases when $R$ is the ring of integers modulo $n$, the direct product of integral domains, the direct product of Artinine local rings and the direct product of two rings such that one of them is not an integral domain.

Introduction to NeutroRings

The objective of this paper is to introduce the concept of NeutroRings by considering three NeutroAxioms (NeutroAbelianGroup (additive), NeutroSemigroup (multiplicative) and NeutroDistributivity (multiplication over addition)). Several interesting results and examples on NeutroRings, NeutroSubgrings, NeutroIdeals, NeutroQuotientRings and NeutroRingHomomorphisms are presented. It is shown that the 1st isomorphism theorem of the classical rings holds in the class of NeutroRings.

On semireversible rings

In this paper, we introduce and study a new class of rings which we call semireversible rings. A ring [Formula: see text] is called semireversible if for any [Formula: see text] implies there exists a positive integer [Formula: see text] such that [Formula: see text]. We observe that the class of semireversible rings strictly lies between the class of central reversible rings and weakly reversible rings. Some relations are provided between semireversible rings and many other known classes of rings. Some extensions of semireversible rings such as ring of fractions, Dorroh extension, subrings of matrix rings are investigated. Finally, we study semireversible rings via modules over them wherein among other results, we prove that a semireversible left (right) SF-ring is strongly regular.

On proper strong Property (𝒜) for rings and modules

The main purpose of this paper is to introduce and investigate a new class of rings lying properly between the class of [Formula: see text]-rings and the class of [Formula: see text]-rings. The new class of rings, termed the class of [Formula: see text]-rings, turns out to share common characteristics with both [Formula: see text]-rings and [Formula: see text]-rings. Numerous properties and characterizations of this class are given as well as the module-theoretic version of [Formula: see text]-rings is introduced and studied.

On the importance of being primitive

We give a brief survey of primitivity in ring theory and in particular look at characterizations of primitive ideals in the prime spectrum for various classes of rings.

A fresh look into skew inverse Laurent series rings

Skew inverse Laurent series rings have been actively investigated recently, and many interesting results have been obtained in the literature. In this paper, by a fresh look, we obtain further results on skew inverse Laurent series rings. First, we give some radical-theoretic properties of this ring construction, giving a characterization of the Jacobson radical of the skew inverse Laurent series ring where the base coefficient ring R is a 2-primal ring and every minimal prime ideal of R is -ideal. As a consequence of the results in first part, we will be able to add a new class of rings, to the several special classes of rings previously shown satisfying the famous Köthe’s Conjecture. We also characterize when a skew inverse Laurent series ring is clean, and we obtain partial characterizations for it to be semiperfect or semiregular.

Extension of Antoine’s Ring Construction

This article concerns a kind of extension of Antoine’s ring construction which is unit-IFP and Armendariz. Such extensions are also shown to be unit-IFP and Armendariz, by which we may extend the classes of unit-IFP and Armendariz rings.

The special linear group for nonassociative rings

Abstract We extend to arbitrary rings a definition of the octonion special linear group due to Baez. At the infinitesimal level, we get a Lie ring, which we describe over some large classes of rings, including all associative rings and all algebras over a field. As a corollary, we compute all the groups Baez defined.

Covering classes, strongly flat modules, and completions

We study some closely interrelated notions of Homological Algebra: (1) We define a topology on modules over a not-necessarily commutative ring R that coincides with the R-topology defined by Matlis when R is commutative. (2) We consider the class $$ \mathcal {SF}$$ of strongly flat modules when R is a right Ore domain with classical right quotient ring Q. Strongly flat modules are flat. The completion of R in its R-topology is a strongly flat R-module. (3) We prove some results related to the question whether $$ \mathcal {SF}$$ a covering class implies $$ \mathcal {SF}$$ closed under direct limits. This is a particular case of the so-called Enochs’ Conjecture (whether covering classes are closed under direct limits). Some of our results concern right chain domains. For instance, we show that if the class of strongly flat modules over a right chain domain R is covering, then R is right invariant. In this case, flat R-modules are strongly flat.

Rings in which elements are a sum of a central and a unit element

In this paper we introduce a new class of rings whose elements are a sum of a central and a unit element, namely a ring $R$ is called $CU$ if each element $a\in R$ has a decomposition $a = c + u$ where $c$ is central and $u$ is unit. One of the main results in this paper is that if $F$ is a field which is not isomorphic to $\Bbb Z_2$, then $M_2(F)$ is a $CU$ ring. This implies, in particular, that any square matrix over a field which is not isomorphic to $\Bbb Z_2$ is the sum of a central matrix and a unit matrix.

Cohomologically Full Rings

Abstract Inspired by a question raised by Eisenbud–Mustaţă–Stillman regarding the injectivity of maps from ${\operatorname{Ext}}$ modules to local cohomology modules and the work by the third author with Pham, we introduce a class of rings, which we call cohomologically full rings. This class of rings includes many well-known singularities: Cohen–Macaulay rings, Stanley–Reisner rings, F-pure rings in positive characteristics, and Du Bois singularities in characteristics $0$. We prove many basic properties of cohomologically full rings, including their behavior under flat base change. We show that ideals defining these rings satisfy many desirable properties, in particular they have small cohomological and projective dimension. When $R$ is a standard graded algebra over a field of characteristic $0$, we show under certain conditions that being cohomologically full is equivalent to the intermediate local cohomology modules being generated in degree $0$. Furthermore, we obtain Kodaira-type vanishing and strong bounds on the regularity of cohomologically full graded algebras.

Generalized Strong Commutativity Preserving Centralizers of Semiprime Γ- Rings

In this paper, we introduce the concept of generalized strong commutativity (Cocommutativity) preserving right centralizers on a subset of a Γ-ring. And we generalize some results of a classical ring to a gamma ring.

Generalized Weyl algebras and diskew polynomial rings

The aim of the paper is to extend the class of generalized Weyl algebras (GWAs) to a larger class of rings (they are also called GWAs) that are determined by two ring endomorphisms rather than one as in the case of ‘old’ GWAs. A new class of rings, the diskew polynomial rings, is introduced that is closely related to GWAs (they are GWAs under a mild condition). Simplicity criteria are given for GWAs and diskew polynomial rings.

2-AB rings and 2-absorbing rings

Let R be a commutative ring with identity. The notion of 2-absorbing ideal was introduced by A. Badawi (Bull Aust Math Soc 75(3):417–429, 2007), as a generalization of prime ideal. A proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, $$c\in R$$ with $$abc\in I$$, then $$ab\in I$$ or $$ac\in I$$ or $$bc\in I$$. A commutative ring R is called a 2-absorbing ring if every nonzero proper ideal of R is 2-absorbing. It is clear that every prime ideal is a 2-absorbing ideal but the converse is not true in general. D. Bennis and B. Fahid introduced a new class of rings in which every 2-absorbing ideal is prime. They called such a ring, a 2-AB ring (Bennis in Beitr Algebra Geom 59(2):391–396, 2018). They showed that if R is a commutative ring such that the prime ideals of R are comparable and $$P^2=P$$, for every prime ideal P of R then R is 2-AB and they asked if $$P^2=P$$ for every prime ideal in a 2-AB ring. In this paper, we give a positive answer to this question in various classes of rings and we investigate the transfer of the 2-AB (respectively 2-absorbing) property to some ring extensions.

Rings whose injective hulls are dual square free

A module M is called dual-square-free (DSF) if M has no proper submodules A and B with and The class of DSF-modules is closed under direct summands and homomorphic images, and a module M is distributive iff every submodule of M is dual-square-free. In this article we consider certain classes of rings R over which is a DSF-module. For example, if R is a right hereditary ring such that is DSF, then R is right noetherian, right distributive, every right ideal of R is two-sided, and every subfactor of RR is quasi-continuous. Also, if R is semilocal and is a DSF-module with small radical, then R is basic, semiperfect and right self-injective. As an immediate application, if R is a right perfect ring such that is DSF as a right R-module then R is right Artinian. If in addition we assume that either or is a DSF-module then the ring R becomes quasi-Frobenius.

Categorified groupoid-sets and their Burnside ring

We explore the category of internal categories in the usual category of (right) group-sets, whose objects are referred to as categorified group-sets. More precisely, we develop a new Burnside theory, where the equivalence relation between two categorified group-sets is given by a particular equivalence between the underlying categories. We also exhibit some of the differences between the old Burnside theory and the new one. Lastly, we briefly explain how to extend these new techniques and concepts to the context of groupoids, employing the categories of (right) groupoid-sets, aiming by this to give an alternative approach to the classical Burnside ring of groupoids.

UNJ-Rings

In analogy to the elementwise definitions of UU- and UJ-rings, a ring [Formula: see text] is called UNJ if [Formula: see text]. After presenting several characterizations and properties, we consider the UNJ-rings within many well-studied classes of rings. In particular, we examine Dedekind finite rings, 2-primal rings, (semi)regular rings, [Formula: see text]-regular rings and rings that satisfy the identity [Formula: see text]. Finally, we conclude this paper with group UNJ-rings.

The McCoy Condition on Skew Poincaré–Birkhoff–Witt Extensions

In this paper, we study the notion of McCoy ring over the class of non-commutative rings of polynomial type known as skew Poincare–Birkhoff–Witt extensions. As a consequence, we generalize several results about this notion considered in the literature for commutative rings and Ore extensions.

Universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras

Let k be a field of arbitrary characteristic, let Λ be a finite dimensional k-algebra, and let V be a finitely generated Λ-module. F. M. Bleher and the third author previously proved that V has a well-defined versal deformation ring R(Λ,V). If the stable endomorphism ring of V is isomorphic to k, they also proved under the additional assumption that Λ is self-injective that R(Λ,V) is universal. In this paper, we prove instead that if Λ is arbitrary but V is Gorenstein-projective then R(Λ,V) is also universal when the stable endomorphism ring of V is isomorphic to k. Moreover, we show that singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over Gorenstein algebras. We also provide examples. In particular, if Λ is a monomial algebra in which there is no overlap (as introduced by X. W. Chen, D. Shen and G. Zhou) we prove that every finitely generated indecomposable Gorenstein-projective Λ-module has a universal deformation ring that is isomorphic to either k or to k〚t〛/(t2).

Commutative rings in which every pure ideal is projective

In this paper, we study the class of rings in which every pure ideal is projective. We investigate the stability of this property under homomorphic image, and its transfer to various contexts of constructions such as pullbacks trivial ring extensions and amalgamation of rings. Our results generate examples which enrich the current literature with new and original families of rings that satisfy this property.