ANNIHILATING SUBMODULE GRAPHS FOR MODULES OVER COMMUTATIVE RINGS

In this article, we give several generalizations of the concept of annihilating an ideal graph over a commutative ring with identity to modules. We observe that, over a commutativering, R, AG (RM) is connected, and diamAG (RM) ≤ 3. More- over, if AG (RM) contains a cycle, then grAG (RM) ≤ 4. Alsofor an R-module M with A (M) ̸= S(M) {0}, A (M) = ∅, 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 sub-module