Annihilator condition on modules

Abstract

Let R be a commutative ring with 1 6¼ 0 and M a unital R-module. M is said to satisfy Property (A) if for each finitely generated ideal J of R contained in ZRðMÞ, there exists 0 6¼ m 2 M such that Jm ¼ ð0Þ: Also M is said to satisfy Property (T) if for each finitely generated submodule N of M contained in TðMÞ; there exists 0 6¼ a 2 R such that aN ¼ ð0Þ: In this article, we study certain annihilator conditions on modules such as Property ðAÞ and Property ðTÞ: In addition to give many properties of modules satisfying Property ðAÞ (Property ðTÞÞ; we characterize these classes of modules in terms of rsubmodules and sr-submodules. Also, we give a method to construct non Noetherian rings in which every ideal satisfies Property (A).