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If M is a torsion-free module over an integral domain, then we show that for each submodule N of M the envelope EM (N ) of N in M is an essential extension of N. In particular, if N is divisible then EM (N ) = N . The last condition says that N is a semiprime submodule of M if N is proper.
Let M be a module over a ring R such that for any ideals a, b of R, (a ∩ b)M = aM ∩ bM . If N is an irreducible and weakly semiprime submodule of M , then we prove that (N :R M ) is a prime ideal of R. As a result, we obtain that if p is an irreducible ideal of a ring R such that a2 ⊆ p (a is an ideal of R) ⇒ a ⊆ p, then p is a prime ideal.
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