Modules with annihilation property

Let R be an associative ring with identity. A right R-module MR is said to have Property (A), if each finitely generated ideal I subset of Z(MR) has a nonzero annihilator in MR. Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155(2) (1971) 505-512.] proved that, over a commutative ring, zero-divisor modules have Property (A). We study and construct various classes of modules with Property (A). Following Anderson and Chun [McCoy modules and related modules over commutative rings, Comm. Algebra 45(6) (2017) 2593-2601.], we introduce G-dual McCoy modules and show that, for every strictly totally ordered monoid G, faithful symmetric modules are G-dual McCoy. We then use this notion to give a characterization for modules with Property (A). For a faithful symmetric right R-module MR and a strictly totally ordered monoid G, it is proved that the right R[G]-module M[G]R[G] is primal if and only if MR is primal with Property (A).