# The annihilating-submodule graph of modules over commutative rings

Research paper by **Habibollah Ansari-Toroghy, Shokoufeh Habibi**

Indexed on: **05 Jan '16**Published on: **05 Jan '16**Published in: **Mathematics - Commutative Algebra**

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#### Abstract

Let M be a module over a commutative ring R. In this paper, we continue our
study of annihilating-submodule graph AG(M) which was introduced in (The
Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42
(2014), 3283{3296). AG(M) is a (undirected) graph in which a nonzero submodule
N of M is a vertex if and only if there exists a nonzero proper submodule K of
M such that NK = (0), where NK, the product of N and K, is defined by (N : M)(K
: M)M and two distinct vertices N and K are adjacent if and only if NK = (0).
We obtain useful characterizations for those modules M for which either AG(M)
is a complete (or star) graph or every vertex of AG(M) is a prime (or maximal)
submodule of M. Moreover, we study coloring of annihilating-submodule graphs.