Annihilator conditions on modules over commutative rings

Let R be a commutative ring and M an R-module. Let Z(M) = {r is an element of R| rm = 0 for some nonzero m is an element of M} and T(M) = {m is an element of M | rm = 0 for some nonzero r is an element of R} M satisfies Property A (respectively, Property T) if for each finitely generated ideal I subset of Z(M) (respectively, finitely generated submodule N subset of T(M)) ann (M)(I) not equal 0 (respectively, ann (R)(N) not equal 0). The ring R satisfies Property A if R-R does. We study rings and modules satisfying Property A or Property T. A number of examples are given, many using the method of idealization.