JOURNAL OF ALGEBRAIC SYSTEMS
Year:2017 | Volume:4 | Issue:2 |Papers Count:6

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say 􀀀 (RM), such that when M = R, 􀀀 (RM) coincide with the zero-divisor graph of R. Many well-known results by D. F. Anderson and P. S. Livingston, have been generalized for 􀀀 (RM). We will show that 􀀀 (RM) is connected with diam(􀀀 (RM))  3, and if 􀀀 (RM) contains a cycle, then gr(􀀀 (RM))  4. We will also show that 􀀀 (RM) = ∅ 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 (􀀀 (RM)) = 1 if and only if 􀀀 (RM) is a star graph. Finally, we study the zero-divisor graph of free R-modules.

Let G be a  nite group and Z(G) be the center of G. For a subset A of G, we de ne kG(A), the number of conjugacy classes of G that intersect A non-trivially. In this paper, we ver-ify the structure of all  nite groups G which satisfy the property kG(G 􀀀 Z(G)) = 5; and classify them.

In this paper, we introduce the notion of fuzzy obsti-nate 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 relation-ships between fuzzy obstinate ideals and the other fuzzy ideals of an MV-algebra. We describe the transfer principle for fuzzy ob-stinate ideals in terms of level subsets. In addition, we show that if  is a fuzzy obstinate ideal of A such that  (0) 2 [0; 1=2], then A= is a Boolean algebra. Finally, we de ne the notion of a normal fuzzy obstinate ideal and investigate some of its properties.

In this paper, the notion of the radical of a  lter in residuated lattices is de ned and several characterizations of the radical of a  lter are given. We show that if F is a positive im-plicative  lter (or obstinate  lter), then Rad(F) = F. We proved the extension theorem for radical of  lters in residuated lattices. Also, we study the radical of  lters in linearly ordered residuated lattices.

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.

Let G = (V (G); E(G)) be a simple connected graph with vertex set V (G) and edge set E(G). The ( rst) edge-hyper Wiener index of the graph G is de ned as: WWe(G) = Σ ff; gg E(G) (de(f; gjG) + d2 e(f; gjG)) = 1 2 Σ f2E(G) (de(fjG) + d2 e (fjG)); where de(f; gjG) denotes the distance between the edges f = xy and g = uv in E(G) and de(fjG) = Σ g2E(G) de(f; gjG). In this pa-per, we use a method, which applies group theory to graph theory, to improving mathematically computation of the ( rst) edge-hyper Wiener index in certain classes of graphs. We give also upper and lower bounds for the ( rst) edge-hyper Wiener index of a graph in terms of its size and Gutman index. Our aim in last section is to investigate products of two or more graphs, and compute the second edge-hyper Wiener index of the some classes of graphs.