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 ⊆ 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).

中文翻译：

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具有湮灭属性的模块

让R是一个与身份相关的环。一个权利R-模块米R据说有财产（一种)，如果每个有限生成的理想一世 ⊆ Z(米R)有一个非零湮没子米R. Evans [类诺特环中的零除数，反式。阿米尔。数学。社会党。 155(2)(1971) 505–512.] 证明了，在交换环上，零除数模块具有属性 (一种）。我们研究和构建了各种类型的具有属性的模块（一种）。继 Anderson 和 Chun [McCoy 模块和交换环上的相关模块之后，通讯。代数 45(6) (2017) 2593–2601.]，我们介绍G-双 McCoy 模块并表明，对于每个严格完全有序的幺半群G, 忠实的对称模块是G-双麦考伊。然后我们使用这个概念来描述具有属性的模块（一种）。对于忠实的对称权利R-模块米R和一个严格完全有序的幺半群G, 证明对R[G]-模块米[G]R[G]是原始的当且仅当米R是原始的属性 (一种）。