New Class Of Rings Research Articles

Rings additively generated by periodic elements

In the this paper, as a generalization of the classical periodic rings, we explore those rings whose elements are additively generated by two (or more) periodic elements by calling them additively periodic. We prove that, in some major cases, additively periodic rings remain periodic too. This includes, for instance, algebraic algebras, group rings, and matrix rings over commutative rings. Moreover, we also obtain some independent results for the new class of rings. For example, the triangular matrix rings retain that property.

( g,e ) -Symmetric Rings

Let [Formula: see text] be a ring and [Formula: see text], [Formula: see text] in [Formula: see text], the set of idempotents of [Formula: see text]. Then [Formula: see text] is called [Formula: see text]-symmetric if [Formula: see text] implies [Formula: see text] for any [Formula: see text], [Formula: see text], [Formula: see text]. Clearly, [Formula: see text] is an [Formula: see text]-symmetric ring if and only if [Formula: see text] is a [Formula: see text]-symmetric ring; in particular, [Formula: see text] is a symmetric ring if and only if [Formula: see text] is a [Formula: see text]-symmetric ring. We show that [Formula: see text] is left semicentral if and only if [Formula: see text] is a [Formula: see text]-symmetric ring; in particular, [Formula: see text] is an Abel ring if and only if [Formula: see text] is a [Formula: see text]-symmetric ring for each [Formula: see text]. We also show that [Formula: see text] is [Formula: see text]-symmetric if and only if [Formula: see text], [Formula: see text] is symmetric, and [Formula: see text] for any [Formula: see text], [Formula: see text]. Using [Formula: see text]-symmetric rings, we construct some new classes of rings.

On maximal ideals of the polynomial ring and some conjectures on Ext-index of rings

Recall that a ring [Formula: see text] is a Hilbert ring if any maximal ideal of [Formula: see text] contracts to a maximal ideal of [Formula: see text]. The main purpose of this paper is to characterize the prime ideals of a commutative ring [Formula: see text] which are traces of the maximal ideals of the polynomial ring [Formula: see text]. In this context, we prove that if [Formula: see text] is a prime ideal of [Formula: see text] such that [Formula: see text] is a semi-local domain of (Krull) dimension [Formula: see text], then [Formula: see text] is the trace of a maximal ideal of [Formula: see text]. Whereas, if [Formula: see text] is Noetherian and either ([Formula: see text]) or (the quotient field of [Formula: see text] is algebraically closed, [Formula: see text] and [Formula: see text] is not semi-local), then [Formula: see text] is never the trace of a maximal ideal of [Formula: see text]. Putting these results into use in investigating the Ext-index of Noetherian rings, we establish connections between the finiteness of the Ext-index of localizations of the polynomial rings [Formula: see text] and the finiteness of the Ext-index of localizations of [Formula: see text]. This allows us to provide a new class of rings satisfying some known conjectures on Ext-index of Noetherian rings as well as to build bridges between these conjectures.

Involution t–Clean Rings with Applications

A new class of rings are introduced which every element in the ring are sum of involution and tripotent elements. This class called involution t-clean ring which is a generalization of invo-clean rings and subclass of clean rings. Some properties of this class are investigate. For an application in graph theory, new graph is defined called t-clean graph of involution t-clean ring the set of vertices is order pairs of involution and tripotent element which is the sum of them is involution t-clean element. The two vertices are adjacent if and only if the sum of involution elements are zero or the product of the tripotent elements are zero. The graphs are connecting, has diameter one and girth three.

Trace ideals of semidualizing modules and two generalizations of nearly Gorenstein rings

In this article, we study the trace ideal of semidualizing modules. Also, we introduce two new classes of rings, namely quasi nearly Gorenstein and weakly nearly Gorenstein, which are generalizations of nearly Gorenstein rings. Furthermore, we show that the trace ideal of a semidualizing module is annihilators of certain Ext modules. Moreover, we give new versions of Tachikawa conjecture using quasi and weakly nearly Gorenstein rings.

Unit lifting morphisms

In this paper, we introduce the concept of a unit lifting morphism, a natural generalization of the classical notion “local morphism”, by providing several examples and investigating all its properties and interrelations with local morphisms and unit lifting ideals. Moreover, we expose a relation between a unit lifting morphism f:R→S of rings and the pair (ker(f),f−1(U(S))) and show that unit lifting morphisms correspond to the least element of a partially ordered set. As a prominent result, we provide the equivalent conditions on the existence of a unit lifting morphism from a ring R into a semisimple artinian ring which is intimately related to a deep result by Camps and Dicks that characterizes semilocal rings in terms of local morphisms. This result gives rise to the study of a new class of rings, which we call weakly semilocal rings.

Connected sums of graded Artinian Gorenstein algebras and Lefschetz properties

In their paper [1], H. Ananthnarayan, L. Avramov, and W.F. Moore introduced a connected sum construction for local Gorenstein rings A,B over a local Gorenstein ring T, which, in the graded Artinian case, can be viewed as an algebraic analogue of the topological construction of the same name. We give two alternative descriptions of this algebraic connected sum: the first uses algebraic analogues of Thom classes of vector bundles and Gysin homomorphisms, the second is in terms of Macaulay dual generators. We also investigate the extent to which the connected sum of A,B over an Artinian Gorenstein algebra T preserves the weak or strong Lefschetz property, thus providing new classes of rings which satisfy these properties.

On the lattice of annihilator ideals and its applications

In this paper we study the properties of (the set of right annihilator ideals of R) as a lattice. We prove that is a Boolean algebra if and only if R is a semiprime ring. For a quasi Armandariz ring R, we show that the lattices and are isomorphic. In general, for any ring R, we show that the lattices and are isomorphic. We answer two questions related to property (a.c.) that were raised by Hong, Kim, Lee and Nielsen. To do this, we introduce a new class of rings. We say R is right strongly quasi Armandariz (-Armandariz) if for each there exists such that The class of right -Armandariz rings is between right -Baer rings and Armandariz rings. For a right -Armandariz ring R, it is proved that R has the right annihilator condition (a.c.) if and only if R[x] has right the annihilator condition (a.c.). We conclude that if R is a quasi Armandariz ring and has right (a.c.), then R[x] has right (a.c.).

On -piecewise Noetherian rings

The purpose of this paper is to introduce two new classes of rings that is closely related to the classes of piecewise Noetherian domains and piecewise w-Noetherian domains. Let is a commutative ring and is a divided prime ideal of Let be a ring with total quotient ring T(R), and set such that for every and every Then is a ring homomorphism from T(R) into and restricted to R is also a ring homomorphism from R into given by for every At this setting, we introduce the notions of -piecewise Noetherian rings and -piecewise w-Noetherian rings and we show that the theories of -piecewise Noetherian (resp. -piecewise w-Noetherian) rings resemble those of piecewise Noetherian (resp. piecewise w-Noetherian) domains.

A GENERALIZATION OF THE THEORY OF STANDARDLY STRATIFIED ALGEBRAS I: STANDARDLY STRATIFIED RINGOIDS

AbstractWe extend the classical notion of standardly stratified k-algebra (stated for finite dimensional k-algebras) to the more general class of rings, possibly without 1, with enough idempotents. We show that many of the fundamental results, which are known for classical standardly stratified algebras, can be generalized to this context. Furthermore, new classes of rings appear as: ideally standardly stratified and ideally quasi-hereditary. In the classical theory, it is known that quasi-hereditary and ideally quasi-hereditary algebras are equivalent notions, but in our general setting, this is no longer true. To develop the theory, we use the well-known connection between rings with enough idempotents and skeletally small categories (ringoids or rings with several objects).

Characterizations of Gelfand Rings Specially Clean Rings and their Dual Rings

In this paper, new criteria for zero dimensional rings, Gelfand rings, clean rings and mp-rings are given. A new class of rings is introduced and studied, we call them purified rings. Specially, reduced purified rings are characterized. New characterizations for pure ideals of reduced Gelfand rings and mp-rings are provided. It is also proved that if the topology of a scheme is Hausdorff, then the affine opens of that scheme is stable under taking finite unions. In particular, every compact scheme is an affine scheme.

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.

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.

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.

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.

Computing the invariants of intersection algebras of principal monomial ideals

We continue the study of intersection algebras [Formula: see text] of two ideals [Formula: see text] in a commutative Noetherian ring [Formula: see text]. In particular, we exploit the semigroup ring and toric structures in order to calculate various invariants of the intersection algebra when [Formula: see text] is a polynomial ring over a field and [Formula: see text] are principal monomial ideals. Specifically, we calculate the [Formula: see text]-signature, divisor class group, and Hilbert–Samuel and Hilbert–Kunz multiplicities, sometimes restricting to certain cases in order to obtain explicit formulæ. This provides a new class of rings where formulæ for the [Formula: see text]-signature and Hilbert–Kunz multiplicity, dependent on families of parameters, are provided.

Rings Whose Elements are Sums of Three or Differences of Two Commuting Idempotents

We define and examine a new class of rings whose elements are the sum of three commuting idempotents or the difference of two commuting idempotents. We fully describe them up to an isomorphism and our obtained results considerably extend some well-known achievements due to Hirano and Tominaga (Bull Aust Math Soc 37:161–164, 1988), to Ying et al. (Can Math Bull 59:661–672, 2016) and to Tang et al. (Lin Multilin Algebra, 2018).

M-Formally Noetherian/Artinian rings

The purpose of this paper is to introduce two new classes of rings that are closely related to the classes of Noetherian rings and Artinian rings. Let R be a commutative unitary ring and m a positive integer. We call R to be m-formally Noetherian (respectively m-formally Artinian) if for every increasing (respectively decreasing) sequence $$(I_n)_{n\ge 0}$$ of ideals (respectively proper ideals) of R, the increasing (respectively decreasing) sequence $$(\sum _{i_1+\cdots +i_m=n}I_{i_1} \ldots I_{i_m})_{n\ge 0}$$ stabilizes. We show that many properties of Noetherian (respectively Artinian) rings are also true for m-formally Noetherian (respectively m-formally Artinian) rings and we give many examples of m-formally Noetherian/Artinian rings. We investigate the m-formally variant of some well known theorems on Noetherian and Artinian rings. We prove that R is m-formally Artinian for some m if and only if R is $$m'$$ -formally Noetherian for some $$m'$$ with zero Krull dimension. We show that the m-formally variant of Eakin-Nagata theorem is true in some way in the zero-dimensional case.