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
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
3Department of Mathematics, Shahid Beheshti University, Evin, Tehran 19834, Iran
Received 11 November 2000
Copyright © 2001 M. R. Pournaki and M. Tousi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.