Journal of Algebraic Systems
Volume:4 Issue: 2, Winter- Spring 2017

ýLet $G$ be a finite group and $Z(G)$ be the center of $G$ý. ýFor a subset $A$ of $G$ý, ýwe define $k_G(A)$ý, ýthe number of conjugacy classes of $G$ which intersect $A$ non-triviallyý. ýIn this paperý, ýwe verify the structure of all finite groups $G$ which satisfy the property $k_G(G-Z(G))=5$ and classify themý.

In this paper, we introduce the notion of fuzzy obstinate ideals in MV -algebras. Some properties of fuzzy obstinate ideals are given. Not only we give some characterizations of fuzzy obstinate ideals, but also bring the extension theorem of fuzzy obstinate ideal of an MV -algebra A. We investigate the relationships between fuzzy obstinate ideals and the other fuzzy ideals of an MV -algebra. We describe the transfer principle for fuzzy obstinate ideals in terms of level subsets. In addition, we show that if \mu is a fuzzy obstinate ideal of A such that \mu(0)\in [0; 1/2], then A/\mu is a Boolean algebra. Finally, we defi ne the notion of a normal fuzzy obstinate ideal and investigate some properties.

ýIn this paperý, ýthe notion of the radical of a filter iný ýresiduated lattices is defined and several characterizations ofý ýthe radical of a filter are givený. ýWe show that if F is aý ýpositive implicative filter (or obstinate filter)ý, ýthený ýRad(F)=Fý. ýWe proved the extension theorem for radical of filters in residuated latticesý. ýAlsoý, ýwe study the radicalý ýof filters in linearly ordered residuated latticesý.

In this paper the notion of Rees short exact sequence for S-posets is introduced, and we investigate the conditions for which these sequences are left or right split. Unlike the case for S-acts, being right split does not imply left split. Furthermore, we present equivalent conditions of a right S-poset P for the functor Hom(P;-) to be exact.

ýLet $G=(V(G),E(G))$ be a simple connected graph with vertex set $V(G)$ and edgeý ýset $E(G)$ý. ýThe (first) edge-hyper Wiener index of the graph $G$ is defined asý: ý$$WW_{e}(G)=\sum_{\{f,g\}\subseteq E(G)}(d_{e}(f,g|G)_{e}^{2}(f,g|G))=\frac 1}{2}\sum_{f\in E(G)}(d_{e}(f|G)^{2}_{e}(f|G)),$$ý ýwhere $d_{e}(f,g|G)$ denotes the distance between the edges $f=xy$ and $g=uv$ in $E(G)$ and $d_{e}(f|G)=\sum_{g\in E(G)}d_{e}(f,g|G)$ý. ýIn this paper we use a methodý, ýwhich applies group theory to graph theoryý, ýto improvingý ýmathematically computation of the (first) edge-hyper Wiener index in certain graphsý. ýWe give also upper and lower bounds for the (first) edge-hyper Wiener index of a graph in terms of its size and Gutman indexý. ýAlso we investigate products of two or more graphs and compute the second edge-hyper Wiener index of the some classes of graphsý. ýOur aim in last section is to find a relation between the third edge-hyper Wiener index of a general graph and the hyper Wiener index of its line graphý. of two or more graphs and compute edge-hyper Wiener number of some classes of graphsý.

Let $R$ be a commutative ring with identity and $M$ an $R$-module. In this paper, we associate a graph to $M$, say ${\Gamma}({}_{R}M)$, such that when $M=R$, ${\Gamma}({}_{R}M)$ coincide with the zero-divisor graph of $R$. Many well-known results by D.F. Anderson and P.S. Livingston have been generalized for ${\Gamma}({}_{R}M)$. We show that ${\Gamma}({}_{R}M)$ is connected with ${\diam}({\Gamma}({}_{R}M))\leq 3$ and if ${\Gamma}({}_{R}M)$ contains a cycle, then $\gr({\Gamma}({}_{R}M))\leq 4$. We also show that ${\Gamma}({}_{R}M)=\emptyset$ if and only if $M$ is a prime module. Among other results, it is shown that for a reduced module $M$ satisfying DCC on cyclic submodules, $\gr{\Gamma}({}_{R}M)=\infty$ if and only if ${\Gamma}({}_{R}M)$ is a star graph. Finally, we study the zero-divisor graph of free $R$-modules.