A remark on prime ideals

Abstract

If M is a torsion-free module over an integral domain, then we show that for each submodule N of M the envelope Eм(N) of N in M is an essential extension of N. In particular, if N is divisible then Eм(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 o f M, then we prove that (N :ʀ 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.