International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 10, Pages 641-643
doi:10.1155/S0161171201011085
A note on the countable union of prime submodules
M. R. Pournaki1
and M. Tousi2
1Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11365-9415, Tehran, Iran
2School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran
Abstract
Let M be a finitely-generated module over a Noetherian ring R. Suppose 𝔞 is an ideal of R and let N=𝔞M and 𝔟=Ann(M/N). If 𝔟⫅J(R), M is complete with respect to the 𝔟-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.