Annihilating Graph of Abelian Groups

Abstract

In [18], the author associated a graph to an R -module M which is precisely a generalization of annihilating ideal graph of a commutative ring, see [15] and [16]. Inasmuch as Abelian groups are precisely Z-modules, in this paper we relate an annihilating graph to an Abelian group , denoted by G(M) , and study this graph. We show that G(M) is an empty graph if and only if either M≅Z or M is a simple Abelian group. Moreover, we show that G(M) is a finite graph if and only if M is a finite Abelian group. Among other things, we characterize Abelian groups for which their annihilating graphs are complete, bipartite or complete bipartite graphs.

Keywords

• modules
• abelian groups
• annihilating graph
• graph