Related papers: On SAP-rings
In this work we attempt to generalize our result in [6] [7] for real rings (not just von Neumann regular real rings). In other words we attempt to characterize and construct real closure * of commutative unitary rings that are real. We also…
Let $R$ be a ring and $S$ a multiplicative subset of $R$. We introduce and study the notions of ($u$-)$S$-$w$-Noetherian modules and ($u$-)$S$-$w$-principal ideal modules. Some characterizations of these new concepts are given.
In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to…
We survey recent progress on the realization problem for von Neumann regular rings, which asks whether every countable conical refinement monoid can be realized as the monoid of isoclasses of finitely generated projective right $R$-modules…
We define and explore in details the class of GUSC rings, that are those rings whose non-invertible elements are uniquely strongly clean. These rings are a common generalization of the so-called USC rings, introduced by Chen-Wang-Zhou in J.…
{Generalizing the notion of nil cleanness from \cite{D13}, in parallel to \cite{DM14}, we define the concept of {\it weak nil cleanness} for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition…
We introduce the notion of pure extending modules, a refinement of classical extending modules in which only pure submodules are required to be essential in direct summands. Fundamental properties and characterizations are established,…
The aim of this paper is to introduce a new class of Noetherian rings of positive characteristic in terms of perfect closures and study their basic properties. If the perfect closure of a Noetherian ring is coherent, we call it an…
We provide two new formulations of the separativity problem. First, it is known that separativity (and strong separativity) in von Neumann regular (and exchange) rings is tightly connected to unit-regularity of certain kinds of elements. By…
We describe rings over which every right module is almost injective. We give a description of rings over which every simple module is a almost projective.
Let $R$ be a commutative ring with $1\neq 0$ and $n$ be a fixed positive integer. A proper ideal $I$ of $R$ is said to be an \textit{$n$-OA ideal} if whenever $a_1a_2\cdots a_{n+1}\in I$ for some nonunits $a_1,a_2,\ldots,a_{n+1}\in R$, then…
Baer's Criterion for Injectivity is a basic tool of the theory of modules and complexes of modules. Its dual version (DBC) is known to hold for all right perfect rings, but its validity for non-right perfect rings is a complex problem…
We present five open problems in the theory of vertex rings. They cover a variety of different areas of research where vertex rings have been, or are threatening to be, relevant. They have also been chosen because I personally find them…
In this paper, we introduce the concept of a "von Neumann regular $\mathcal{C}^{\infty}$-ring", which is a model for a specific equational theory. We delve into the characteristics of these rings and demonstrate that each Boolean space can…
We propose definitions of SVD, spectral decomposition (for self-adjoint matrices) and Jordan decomposition which make sense for all rings. For many rings, these decompositions can be shown to exist. For some specific rings, these…
In this note, we give several characterizations of left pure-semisimple in terms of the (pre)envelope, (pre)cover, direct limits, direct sums, inverse limits and direct products properties of pure-projective modules or pure-injective…
An associative ring $R$ with identity is left pseudo-morphic if for every $a$$\in$$R$, there exists $b$$\in$$R$ such that $Ra=l_R(b)$. If, in addition, $l_R(a)=Rb$, then $R$ is called left morphic. $R$ is morphic if it is both left and…
We investigate the notion of \textit{semi-nil clean} rings, defined as those rings in which each element can be expressed as a sum of a periodic and a nilpotent element. Among our results, we show that if $R$ is a semi-nil clean NI ring,…
Both the classes of $R$-coneat injective modules and its superclass, pure Baer injective modules, are shown to be preenveloping. The former class is contained in another one, namely, self coneat injectives, i.e. modules $M$ for which every…
We define the class of {\it CUSC} rings, that are those rings whose clean elements are uniquely strongly clean. These rings are a common generalization of the so-called {\it USC} rings, introduced by Chen-Wang-Zhou in J. Pure \& Applied…