ISOMORPHISMS ON ANNIHILATOR GRAPH OF MODULES

Let R be a commutative ring with identity, and M be an R-module. In [1], we focus on annihilator graph of modules, AG(M), the set T(M)* is a non-zero torsion elements of M —would be the vertices of annihilator graph of modules, and x, y ∈ T(M)* were adjacent if and only if AnnR ([x M]y)≠ AnnR (x) ∪ AnnR (y) or AnnR ([y : M]x) ≠ AnnR (x) ∪ AnnR (y). We investigate the structure, the diameter, and the girth of annihilator graph of R-modules [1]. Ghalandarzadeh and Malakooti Rad in [12] proved that for torsion graph of an R-module M is Γ(M) whose vertices are nonzero torsion elements of M, and two distinct vertices x and y are adjacent if and only if [x : M][y : M]M = {0???? }, if S = R (M), then Γ(M) and Γ(???? −1M) are isomorphic for a multiplication R-module M. The purpose of this paper is to study the connection between the AG(M) and AG(???? −1M). We show that if S = R (M), then AG(M) and AG(???? −1M) are isomorphic for a multiplication R-module M.