Recall that a ring $R\ $is said to be quasi regular if its total quotient $q(R)\ \textit{von Neumann regular}. It is well known and only it reduced satisfying the property: for each $a\in R,$ $ann_{R}(ann_{R}(a))=ann_{R}(b)$ some $b\in R$. Here, in this study, we extend notion of rings which satisfy aforementioned property modules. We give many characterizations properties these two classes Mor...

let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

In this article, we realize the finite range ultragraph Leavitt path algebras as Steinberg algebras. This realization allows us to use groupoid approach obtain structural results about these Using skew product of groupoids, show that are graded von Neumann regular rings. We characterize strongly and every algebra is semiprimitive. Moreover, irreducible representations also can be realized Cuntz...

In our earlier study [LR] of prime ideal principles in commutative rings, we have introduced the notion of Oka and Ako families of ideals, along with their “strong analogues”. The logical hierarchy between these ideal families (and the classically well-known monoidal families of ideals) was partly worked out in [LR] over general commutative rings. In the present paper, we amplify this study by ...

We showed special types of comprehensive Gröbner bases can be defined and calculated as the applications of Gröbner bases in polynomial rings over commutative Von Neumann regular rings in [5] and [6]. We called them discrete comprehensive Gröbner bases, since there is a strict restriction on specialization of parameters, that is parameters can take values only 0 and 1. In this paper, we show th...

Abstract A module M is called $$\mathfrak {s}$$ s -coseparable if for every nonzero submodule U of such that / finitely generated, there exists a direct summand V $$V \subseteq U$$ V ⊆ U and generated. It shown non-finitely generated fr...

Let $R$ be an associative ring with identity. An element $x in R$ is called $mathbb{Z}G$-regular (resp. strongly $mathbb{Z}G$-regular) if there exist $g in G$, $n in mathbb{Z}$ and $r in R$ such that $x^{ng}=x^{ng}rx^{ng}$ (resp. $x^{ng}=x^{(n+1)g}$). A ring $R$ is called $mathbb{Z}G$-regular (resp. strongly $mathbb{Z}G$-regular) if every element of $R$ is $mathbb{Z}G$-regular (resp. strongly $...

A ring $R$ is called right CSP if the sum of any two closed ideals also a ideal $R$. Left rings can be defined similarly. An example given to show that left may not CSP. It shown matrix over proved $\mathbb{M}_{2}(R)$ and only self-injective von Neumann regular. The equivalent characterization for trivial extension $R\propto R$

We study topological von Neumann regularity and principal von Neumann regularity of Banach algebras. Our main objective is comparing these two types of Banach algebras and some other known Banach algebras with one another. In particular, we show that the class of topologically von Neumann regular Banach algebras contains all $C^*$-algebras, group algebras of compact abelian groups and ...