Associated Prime Submodules of Finitely Generated Modules

Abstract

ABSTRACT Let R be a commutative ring with identity. For a finitely generated R-module M, the notion of associated prime submodules of M is defined. It is shown that this notion inherits most of the essential properties of the usual notion of associated prime ideals. In particular, it is proven that for a Noetherian multiplication module M, the set of associated prime submodules of M coincides with the set of M-radicals of primary submodules of M which appear in a minimal primary decomposition of the zero submodule of M. Also, Anderson's (1994 Anderson , D. D. ( 1994 ). A note on minimal prime ideals . Proc. Amer. Math. Soc. 122 : 13 – 14 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) theorem is extended to minimal prime submodules in a certain type of modules.