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Let M be a finitely-generated module over a Noetherian ring R. Suppose  is an ideal of R and let N=M and =Ann(M/N). If ⫅J(R), M is complete with respect to the -adic topology, {Pi}i≥1 is a countable family of prime submodules of M, and x∈M, then x+N⫅∪i≥1Pi implies that x+N⫅Pj for some i≥1. This extends a theorem of Sharp and Vámos concerning prime ideals to prime submodules

A note on the countable union of prime submodules