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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 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 $M$-radicals 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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