Related papers: Rings additively generated by idempotents and nilp…

Let R be a 2 torsion free semiprime ring and d a nonzero derivation. Further let A = O(R) be the orthogonal completion of R and B = B(C) the Boolean ring of C where C be the extended centroid of R. We show that if a[[d(x),x]^n- [y,…

We provide a new proof that the upper nilradical of a compact ring coincides with the sum of its left nil ideals using the properties of orthogonal idempotents in compact rings.

In this paper, we say a ring $R$ is Nil$_{\ast}$-Noetherian provided that any nil ideal is finitely generated. First, we show that the Hilbert basis theorem holds for Nil$_{\ast}$-Noetherian rings, that is, $R$ is Nil$_{\ast}$-Noetherian if…

In \cite{C, LCC}, it is proven that if an element $ab$ in a ring is (generalized) Drazin invertible, then so is $ba$. In this paper, we give a new and short proof of it in an effective manner. In particular, we show that if $ab$ is strongly…

Let $R$ be a finite unitary ring such that $R=R_0[R^*]$ where $R_0$ is the prime ring and $R^*$ is not a nilpotent group. We show that if all proper subgroups of $R^*$ are nilpotent groups, then the cardinal of $R$ is a power of prime…

We show that a suitable ring with a ``nice'' topology, in which convergent limits of units are units, is an \aleph_0-exchange ring. We generalize the argument to show that a semi-regular ring, R, with a ``nice'' topology, is a full exchange…

All rings considered are commutative. In this article we introduce and study two notions of modules which are stronger than CS modules, namely weakly IN modules and strongly CS modules. Our main aim is to characterize when a trivial…

Let $R$ be a ring with unity. The \emph{idempotent graph} $G_{\text{Id}}(R)$ of a ring $R$ is an undirected simple graph whose vertices are the set of all the elements of ring $R$ and two vertices $x$ and $y$ are adjacent if and only if…

A well-known theorem of Wedderburn asserts that a finite division ring is commutative. In a division ring the group of invertible elements is as large as possible. Here we will be particularly interested in the case where this group is as…

We introduce the class of sober rings and investigate it through several key results, highlighting connections to some other known classes of rings. We analyze sufficient conditions for a ring to be sober, as well as necessary conditions.…

If R is a commutative ring, we prove that every finitely generated module has a pure-composition series with indecomposable factors and any two such series are isomorphic if and only if R is a Bezout ring and a CF-ring.

This article introduces the notion of an NJ-reflexive ring and demonstrates that it is distinct from the concept of a reflexive ring. The class of NJ-reflexive rings contains the class of semicommutative rings, the class of left (right)…

We use George Bergman's recent normal form for universally adjoining an inner inverse to show that, for general rings, a nilpotent regular element $x$ need not be unit-regular. This contrasts sharply with the situation for nilpotent regular…

The separability tensor element of a separable extension of noncommutative rings is an idempotent when viewed in the correct endomorphism ring; so one speaks of a separability idempotent, as one usually does for separable algebras. It is…

This paper presents an extension of the concept of NR-clean introduced in [12] to graded ring theory. We define and explore graded NR-clean rings, which generalize the class of graded U-nil clean previously studied in [15]. We provide…

In this paper, we are mainly interested in the two questions "which are the commutative rings on which every finitely presented modules is [Formula: see text]-periodic (respectively, [Formula: see text]-periodic)?". It is proved that these…

We give necessary and sufficient conditions for weak and strong embeddability of amalgams in each subvariety of the category of all nilpotent groups of class at most two; this generalizes B. Maier's result for the latter class. We also…

We give a comprehensive study of the so-called \textit{semi-tripotent rings} obtaining their new and non-trivial characterization as well as a complete description in terms of sums and products of some special elements. Particularly, we…

Let R be a commutative ring with identity and N(R) be the set of all nilpotent elements of R. The aim of this paper is to introduce and study the notion of nil-prime ideals as a generalization of prime ideals. We say that a proper ideal P…

In this paper, we define weakly coherent rings, and examine the transfer of these rings to homomorphic image, trivial ring extension, localization, and direct product. These results provide examples of weakly coherent rings that are not…