Journal of Algebraic Structures and Their Applications
Volume:8 Issue: 1, Winter Spring 2021

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the paper.

In this paper, some categorical properties of the category { Pre-Dcpo} of all pre-dcpos; pre-ordered sets which are also pre-directed complete, with pre-continuous maps between them is considered. In particular, we characterize products and coproducts in this category. Furthermore, we show that this category is neither complete nor cocomplete. Also, epimorphisms and monomorphisms in {Pre-Dcpo} are described.Finally, some adjoint relations between the category {Pre-Dcpo} and others are considered.More precisely, we consider the forgetful functors between this category and some well-known categories, and study the existence of their left and right adjoints.

Let $R$ be an associative ring with unity. An element $x in R$ is called $mathbb{Z}G$-clean if $x=e+r$, where $e$ is an idempotent and $r$ is a $mathbb{Z}G$-regular element in $R$. A ring $R$ is called $mathbb{Z}G$-clean if every element of $R$ is $mathbb{Z}G$-clean. In this paper, we show that in an abelian $mathbb{Z}G$-regular ring $R$, the $Nil(R)$ is a two-sided ideal of $R$ and $frac{R}{Nil(R)}$ is $G$-regular. Furthermore, we characterize $mathbb{Z}G$-clean rings. Also, this paper is involved with investigating $mathbb{F}_{2}C_{2}$ as a social group and measuring influence a member of it’s rather than others.

This paper  introduces a novel concept of  Boolean function--based hypergraph  with respect to any given T.B.T(total binary truth table). This study defines a notation of  kernel set   on  switching functions and proves  that every  T.B.T corresponds to a  Minimum   Boolean expression via  kernel set  and presents  some conditions on  T.B.T to obtain  a Minimum irreducible   Boolean expression from switching functions. Finally, we present an algorithm and so Python programming(with complete and original codes) such that for any given T.B.T, introduces a Minimum irreducible   switching expression.

In this article we study and investigate the behavior of $r$-submodules (a proper submodule $N$ of an $R$-module $M$  in which $amin N$ with ${rm Ann}_M(a)=(0)$ implies that $min N$ for each $ain R$ and $min M$). We show that every simple submodule, direct summand,  divisible submodule, torsion submodule and the socle of a module is an $r$-submodule and if $R$ is a domain, then the singular submodule is an $r$-submodule. We also introduce the concepts of $uz$-module (i.e., an $R$-module $M$ such that either ${rm Ann}_M(a)not=(0)$ or $aM=M$, for every $ain R$) and strongly  $uz$-module (i.e., an $R$-module $M$ such that $aMsubseteq a^2M$, for every $ain R$) in the category of modules over commutative rings. We show that every Von Neumann regular module is a strongly $uz$-module and every Artinian  $R$-module is a   $uz$-module. It is observed that if $M$ is a faithful cyclic $R$-module, then  $M$ is a   $uz$-module if and only if every its cyclic submodule is an $r$-submodule. In addition, in this case, $R$ is a domain if and only if the only $r$-submodule of $M$ is zero submodule. Finally, we prove that $R$  is a $uz$-ring if and only if every faithful cyclic $R$-module is a $uz$-module.

This paper introduces and investigates the notion of a generalized Stone residuated lattice. It is observed that a residuated lattice is generalized Stone if and only if it is quasicomplemented and normal. Also, it is proved that a finite residuated lattice is generalized Stone if and only if it is normal. A characterization for generalized Stone residuated lattices is given by means of the new notion of $alpha$-filters. Finally, it is shown that each non-unit element of a directly indecomposable generalized Stone residuated lattice is a dense element.