THE ANNIHILATOR GRAPH OF MODULES OVER COMMUTATIVE RINGS

Let m be a module over a commutative ring r, z∗(m) be its set of weak zero-divisor elements, and if m ∈ m, then let im = (rm :r m) = {r ∈ r : rm ⊆ rm}. the annihilator graph of m is the (undirected) graph ag(m) with vertices z˜ ∗(m) = z∗(m) {0}, and two distinct vertices m and n are adjacent if and only if (0 :r iminm) 6= (0 :r m) ∪ (0 :r n). we show that ag(m) is connected with diameter at most two and girth at most four. also, we study some properties of the zero-divisor graph of reduced multiplication-like r-modules.