ANNIHILATING SUBMODULE GRAPHS FOR MODULES OVER COMMUTATIVE RINGS

Baziar, M.

doi:10.22044/jas.2015.487

ANNIHILATING SUBMODULE GRAPHS FOR MODULES OVER COMMUTATIVE RINGS

Document Type : Original Manuscript

Author

M. Baziar

Department of Mathematics, University of Yasouj, P.O.Box 75914, Yasouj, Iran.

10.22044/jas.2015.487

Abstract

In this article, we give several generalizations of the concept of annihilating ideal graph over a commutative ring with identity to modules. We
observe that over a commutative ring

R

B b b {A G}_{*} (_{R} M)

is
connected and diam

B b b {A G}_{*} (_{R} M) l e q 3

. Moreover, if

B b b {A G}_{*} (_{R} M)

contains a cycle, then

m b o x g r B b b {A G}_{*} (_{R} M) l e q 4

. Also for an

R

-module

M

with

B b b A_{*} (M) n e q S (M) s e t m i n u s 0

B b b A_{*} (M) = e m p t y s e t

if and only if

M

is a uniform module and ann

(M)

is a prime
ideal of

R

Keywords

zero-divisor graph
Annihilating submodule graph
Weakly annihilating submodule

Volume 3, Issue 1 - Serial Number 1
September 2015
Pages 39-47

ANNIHILATING SUBMODULE GRAPHS FOR MODULES OVER COMMUTATIVE RINGS

Volume 3, Issue 1 - Serial Number 1September 2015Pages 39-47

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Volume 3, Issue 1 - Serial Number 1
September 2015
Pages 39-47