for any reduced commutative $f$-ring with identity and bounded inversion, we show that a condition which is obviously necessary for the socle of the ring to coincide with the socle of its bounded part, is actually also sufficient. the condition is that every minimal ideal of the ring consist entirely of bounded elements. it is not too stringent, and is satisfied, for instance, by rings of conti...

In this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $Z_{2}$. We also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.

Let $R$ be a unitary ring with an endomorphism $σ$ and $F∪{0}$ be the free monoid generated by $U={u_1,…,u_t}$ with $0$ added, and $M$ be a factor of $F$ setting certain monomial in $U$ to $0$, enough so that, for some natural number $n$, $M^n=0$. In this paper, we give a sufficient condition for a ring $R$ such that the skew monoid ring $R*M$ is quasi-Armendariz (By Hirano a ring $R$ is called...

The rings considered in this article are commutative rings with identity $1neq 0$. The aim of this article is to define and study the exact annihilating-ideal graph of commutative rings. We discuss the interplay between the ring-theoretic properties of a ring and graph-theoretic properties of exact annihilating-ideal graph of the ring.

Quantum chemical (MNDo)caleulations have beenused toelucidate the molecular properties and structure - activity relationships of dihydropyridine (DHP) type calcium antagonists. There is a good corielation between the net atomic charges on various atoms of the 4 - phenyl ring of dihydropyridines and pharmacological activity. .Also, activity decreases with increasing free rotation of the phen...

let r be a right gf-closed ring with finite left and right gorenstein global dimension. we prove that if i is an ideal of r such that r/i is a semi-simple ring, then the gorensntein flat dimensnion of r/i as a right r-module and the gorensntein injective dimensnnion of r/i as a left r-module are identical. in particular, we show that for a simple module s over a commutative gorensntein ring r, ...

this paper describes the impact of the distance and the size of the corona ring, on the magnitude anddistribution of electrical potential across the polymer insulators. the procedure is based on finiteelement method numerical analysis and stochastic optimization algorithm of differential evolution(de). the optimal selection of corona ring has a significant impact on the control of potential and...

For any reduced commutative $f$-ring with identity and bounded inversion, we show that a condition which is obviously necessary for the socle of the ring to coincide with the socle of its bounded part, is actually also sufficient. The condition is that every minimal ideal of the ring consist entirely of bounded elements. It is not too stringent, and is satisfied, for instance, by rings of ...

In this paper, we define and study the notion of the real-valued functions on a frame $L$. We show that $F(L) $, consisting of all frame homomorphisms from the power set of $mathbb{R}$ to a frame $ L$, is an $f$-ring, as a generalization of all functions from a set $X$ into $mathbb R$. Also, we show that $F(L) $ is isomorphic to a sub-$f$-ring of $mathcal{R}(L)$, the ring of real-valued continu...

This article examines annihilators in the skew polynomial ring $R[x;alpha,delta]$. A ring is strongly right $AB$ if everynon-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property ($A$) and the conditions asked by P.P. Nielsen. We assume that $R$ is an ($alpha$,$delta$)-compatible ring, and prove that, if $R$ is ni...