JOURNAL OF ALGEBRAIC SYSTEMS
Year:2016 | Volume:4 | Issue:1 |Papers Count:7

Let (G; N) be a pair of groups. In this paper,  rstly, we construct a relative central extension for the pair (G; N) such that special types of the covering pair of (G; N) are the homomor-phic image of it. Secondly, we show that every perfect pair admits at least one covering pair. Finally, among extending some proper-ties of perfect groups to perfect pairs, we characterize the covering pairs of a perfect pair (G; N) under some extra assumptions.

Let R be a commutative ring with an identity, and M be a unitary R-module. In this paper, the concept of multiplicatively-closed subset of R is generalized, and some properties of the generalized subsets of M are studied. Some well-known theorems about multiplicatively-closed subsets of R are also generalized, and it is shown that some other well-known results are not valid for M.

Let H be a subgroup of a  nite group G. We say that H is SS-semipermutable in G if H has a supplement K in G such that H permutes with every Sylow subgroup X of K with (jXj; jHj) = 1. In this paper, the structure of SS-semipermutable subgroups and the  nite groups in which SS-semi-permutability is a transitive relation are described. It is shown that a  nite solvable group G is a PST-group if and only if whenever H  K are two p-subgroups of G, H is SS-semipermutable in NG(K).

The prime graph of a  nite group G is denoted by 􀀀 (G). A nonabelian simple group G is called quasirecognizable by prime graph if, for every  nite group H, where 􀀀 (H) = 􀀀 (G), there exists a nonabelian composition factor of H that is isomorphic to G. Until now, it has been proved that some  nite linear simple groups are quasirecognizable by prime graph, for instance, the lin-ear groups Ln(2) and Ln(3) are quasirecognizable by prime graph. In this paper, we consider the quasirecognition by prime graph of the simple group Ln(5).

Let R be a commutative ring. An R-module M is called co-multiplication, provided that for each submodule N of M, there exists an ideal I of R such that N = (0: M I). In this paper, we show that co-multiplication modules are a generaliza-tion of strongly duo modules. Uniserial modules of  nite length, and hence, valuation Artinian rings are some distinguished classes of co-multiplication modules. In addition, if R is a Noetherian quasi-injective ring, then R is strongly duo if and only if R is co-multiplication. We also show that J-semisimple strongly duo rings are precisely semisimple rings. Moreover, if R is a perfect ring, then uniserial R-modules are co-multiplication of  nite length modules. Finally, we show that Abelian co-multiplication groups are all reduced, and co-multiplication Z-modules (Abelian groups) are characterized as well.

In this work, we study the signed Roman domination number of the join of graphs. Specially, we determine it for the join of cycles, wheels, fans, and friendship graphs.

Let R be a commutative Noetherian ring, and let a and b be two ideals of R such that R=(a + b) is Artinian. Let M and N be two  nitely generated R-modules. We prove that Hj b (Ht a (M; N)) is Artinian for j = 0; 1, where t = inffi 2 N0: Hi a(M; N) is not  nitely generatedg. Also, we prove that if dim Supp (Hi a(M; N))  2, then H1 b (Hi a(M; N)) is Artinian for all i. More-over, we show that if dimN = d, then Hj b (Hd􀀀 1 a (N)) is Artinian for all j  1.