Let R be a commutative ring with identity. For a finitely
generated Rmodule M, the notion of associated prime
submodules of M is defined. It is shown that this notion
inherits most of essential properties of the usual notion of
associated prime ideals. In particular, it is proved that for a
Noetherian multiplication module M, the set of associated prime
submodules of M coincides with the set of Mradicals of
primary submodules of M which appear in a minimal primary
decomposition of the zero submodule of M. Also, Anderson's
theorem [2] is extended to minimal prime submodules in a certain
type of modules.
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