Journal of Algebra and Related Topics
Volume:8 Issue: 2, Autumn 2020

‎Let $R$ be a commutative ring with identity‎, ‎$S$ a multiplicatively closed subset of $R$‎, ‎and $M$ be an $R$-module‎. ‎The goal of this work is to introduce the notion of $S$-pure submodules of $M$ as a generalization of pure submodules of $M$ and prove a number of results concerning of this class of modules‎. ‎We say that a submodule $N$ of $M$ is emph {$S$-pure} if there exists an $s in S$ such that $s(N cap IM) subseteq IN$ for every ideal $I$ of $R$‎. ‎Also‎, ‎We say that $M$ is emph{fully $S$-pure} if every submodule of $M$ is $S$-pure‎.

In this paper, we prove that a nonzero square closed $*$-Lie ideal $U$ of a $*$-prime ring $Re$ of Char $Re$ $neq$ $(2^{n}-2)$ is central, if one of the following holds: $(i)delta(x)delta(y)mp xcirc yin Z(Re),$ $(ii)[x,y]-delta(xy)delta(yx)in Z(Re),$ $(iii)delta(x)circ delta(y)mp [x,y]in Z(Re),$ $(iv)delta(x)circ delta(y)mp xyin Z(Re),$ $(v) delta(x)delta(y)mp yxin Z(Re),$ where $delta$ is the trace of $n$-additive map $digamma: underbrace{Retimes Retimes....times Re}_{n-times}longrightarrow Re$,$~mbox{for all}~ x,yin U$.

Let $(sigma,delta)$ be a quasi derivation of a ring $R$ and $M_R$ a right $R$-module. In this paper, we introduce the notion of $(sigma,delta)$-skew McCoy modules which extends the notion of McCoy modules and $sigma$-skew McCoy modules. This concept can be regarded also as a generalization of $(sigma,delta)$-skew Armendariz modules. We study some connections between reduced modules, semicommutative modules, $(sigma,delta)$-compatible modules and $(sigma,delta)$-skew McCoy modules. Furthermore, we will give some results showing that the property of being an $(sigma,delta)$-skew McCoy module transfers well from a module $M_R$ to its skew triangular matrix extensions and vice versa.

The determining number of a graph $G = (V,E)$ is the minimum cardinality of a set $Ssubseteq V$ such that pointwise stabilizer of $S$ under the action of $Aut(G)$ is trivial. In this paper, we compute the determining number of some families of cubic graphs.

This paper introduces and studies the notion of Property (A) of a ring R or an R-module M along an ideal I of R. For instance, any module M over R satisfying the Property (A) do satisfy the Property (A) along any ideal I of R. We are also interested in ideals I which are A-module along themselves. In particular, we prove that if I is contained in the nilradical of R, then any R-module is an A-module along I and, thus, I is an A-module along itself. Also, we present an example of a ring R possessing an ideal I which is an A-module along itself while I is not an A-module. Moreover, we totally characterize rings R satisfying the Property (A) along an ideal I in both cases where I⊆\Z(R) and where I⊈\Z(R). Finally, we investigate the behavior of the Property (A) along an ideal with respect to direct products.

In this paper, we define and consider G-set on Γsemihypergroups and we obtain relations between G-sets and their associate Gb-sets where G is a Γ-semihypergroup and Gb is an associated semihypergroup. Finally, we obtain the relation between direct limit of Gb-sets from the direct limit defined on G-sets.