Related papers: A Note on $*$-Clean Rings

We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl's question whether or not $R$ being nil-clean implies that $\mathbb{M}_n(R)$ is…

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…

In this paper we introduce the notion of "strong $n$-perfect rings" which is in some way a generalization of the notion of "$n$-perfect rings". We are mainly concerned with those class of rings in the context of pullbacks. Also we exhibit a…

Using the notion of cyclically pure injective modules, a characterization for rings which are locally valuation is established. As applications, new characterizations for Prufer domains and pure semi-simple rings are provided. Namely, we…

In this paper, we introduce the concept of graded m-nil clean ring to extend the existing notion of graded nil-clean ring introduced in [10]. We explore fundamental properties of these rings, emphasizing the interplay between the identity…

In this paper, we introduce a class of rings in which every nilpotent element is central. This class of rings generalizes so-called reduced rings. A ring $R$ is called {\it central reduced} if every nilpotent element of $R$ is central. For…

This paper unifies several generalizations of coherent rings in one notion. Namely, we introduce $n$-$\mathscr{X}$-coherent rings, where $\mathscr{X}$ is a class of modules and $n$ is a positive integer, as those rings for which the…

In this article, we show that for a partial skew group ring R*G, where R is a commutative ring, each non-zero ideal of R*G intersects R non-trivially if and only if R is a maximal commutative subring of R*G. As a consequence, we obtain…

If $R$ is a regular and semiartinian ring, it is proved that the following conditions are equivalent: (1) $R$ is unit-regular, (2) every factor ring of $R$ is directly finite, (3) the abelian group $K_0(R)$ is free and admits a basis which…

This paper introduces and studies a new class of rings called {\it $U\sqrt{\Delta}$-rings}. A ring $R$ is $U\sqrt{\Delta}$ if every non-unit element can be written as the product of a unit and an element from $\sqrt{\Delta(R)}$, where…

In this paper, a combination of algebraic and topological methods are applied to obtain new and structural results on harmonic rings. Especially, it is shown that if a Gelfand ring $A$ modulo its Jacobson radical is a zero dimensional ring,…

In this paper, we introduce and study the class $S$-$\mathcal{F}$-ML of $S$-Mittag-Leffler modules with respect to all flat modules. We show that a ring $R$ is $S$-coherent if and only if $S$-$\mathcal{F}$-ML is closed under submodules. As…

Separative von Neumann regular rings exist in abundance. For example, all regular self-injective rings, unit regular rings, regular rings with a polynomial identity are separative. It remains open whether there exists a non-separative…

Let $R$ be an associative ring with identity and let $N$ be a nil ideal of $R$. It is shown that units of $R/N$ can be lifted to units in $R$. Under some mild conditions on the ring, a procedure is given to determine those lifted units in a…

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…

A ring $R$ is called right SSP (SIP) if the sum (intersection) of any two direct summands of $R_{R}$ is also a direct summand. Left sides can be defined similarly. The following are equivalent: (1) $R$ is right SSP. (2) $R$ is right C3 and…

We introduce an analogue to Quasi-$F$-splittings, Quasi-$F$-purity, which is definable over rings that are not necessarily $F$-finite. We show that this property is equivalent to being Quasi-$F$-split in the complete local and $F$-finite…

In this note, we show that a strongly $\phi$-ring $R$ is a $\phi$-PvMR if and only if any $\phi$-torsion free $R$-module is $\phi$-$w$-flat, if and only if any divisible module is nonnil-absolutely $w$-pure module, if and only if any…

We investigate the behavior of four coherent-like conditions in regular conductor squares. In particular, we find necessary and sufficient conditions in order that a pullback ring be a finite conductor ring, a coherent ring, a generalized…

We are working in the category of commutative unital rings and denote by $\mathrm U(R)$ the group of units of a nonzero ring $R$. An extension of rings $R\subseteq S$, satisfying $\mathrm U(R)=R \cap\mathrm U(S)$ is usually called local.…