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 RG-module MGRG is primal if and only if MR is primal with Property (A).
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