相关论文: Centrally Essential Rings and Semirings

Let $R$ be a finite ring and let $\Cent(R)$ denote the set of all distinct centralizers of $R$. $R$ is called an $n$-centralizer ring if $|Cent(R)| = n$. In this paper, we characterize $n$-centralizer finite rings for $n \leq 7$.

A semiring can be ``completed'' (i.e., embedded into a semiring in which all infinite sums are defined and satisfy some reasonable properties) iff this semiring can be naturally partially ordered. This construction is ``natural'' (a left…

It is proved that a ring $A$ is a right or left Noetherian, right distributive centrally essential ring if and only if $A=A_1\times\cdots\times A_n$, where each of the rings $A_i$ is either a commutative Dedekind domain or a uniserial…

Two elements $a,b$ in a ring $R$ form a right coprime pair, written $\langle a,b\rangle$, if $aR+bR=R$. Right coprime pairs have shown to be quite useful in the study of left cotorsion or exchange rings. In this paper, we define the class…

In regard to our recent studies of rings with (strongly, weakly) nil-clean-like properties, we explore in-depth both the structural and characterization properties of those rings whose elements that are not units are weakly nil-clean. Group…

Idempotents yield much insight in the structure of finite semigroups and semirings. In this article, we obtain some results on (multiplicatively) idempotents of the endomorphism semiring of a finite chain. We prove that the set of all…

In this paper, we describe finite, additively commutative, congruence simple semirings. The main result is that the only such semirings are those of order 2, zero-multiplication rings of prime order, matrix rings over finite fields, those…

We define the rank of elements of general unital rings, discuss its properties and give several examples to support the definition. In semiprime rings we give a characterization of rank in terms of invertible elements. As an application we…

We develop the theory of central ideals on commutative rings. We introduce and study the central seminormalization of a ring in another one. This seminormalization is related to the theory of regulous functions on real algebraic varieties.…

Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no non-zero injective semimodules (e.g. the semiring of non-negative integers). In this…

In this paper, we introduce and study a strict generalization of symmetric rings. We call a ring $R \,\,\, 'P-symmetric'$ if for any $a,\, b,\, c\in R,\, abc=0$ implies $bac\in P(R)$, where $P(R)$ is the prime radical of $R$. It is shown…

It is proved that a ring $R$ is a right uniserial, right Noetherian centrally essential ring if and only if $R$ is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. It is also proved that…

In this paper, we introduce and study two new classes of commutative rings, namely semi transitional rings and transitional rings, which extend several classical ideas arising from rings of continuous functions and their variants. A general…

The purpose of this paper is to give a partial positive answer to a question raised by Khurana et al. as to whether a ring $R$ with stable range one and central units is commutative. We show that this is the case under any of the following…

Since for the classification of finite (congruence-)simple semirings it remains to classify the additively idempotent semirings, we progress on the characterization of finite simple additively idempotent semirings as semirings of…

Let R be a semiring. We say that a non-zero subsemimodule S of an R-semimodule M is second if for each a \in R, we have aS = S or aS = 0. The aim of this paper is to study the notion of second subsemimodules of semimodules over commutative…

A ring is called a commutator ring if every element is a sum of additive commutators. In this paper we give examples of such rings. In particular, we show that given any ring R, a right R-module N, and a set X, End_R(\bigoplus_X N) and…

In the set of continuous functions C(X,Y) where Y has a topology close to being discrete, there is an equivalence relation on X which characterizes the quasi-components of X. If Y satisfies weak algebraic conditions with a single binary…

The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring is fully elementary, i.e. we can express symmetric polynomials with elementary ones in a unique way. The…

The main goal of this article is to introduce the concept of $EM-G-$graded rings. This concept is an extension of the notion of $EM-$rings. Let $G$ be a group and $R$ be a $G-$graded commutative ring. The $G-$gradation of $R$ can be…