International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 10, Pages 641-643

A note on the countable union of prime submodules

M. R. Pournaki1,2 and M. Tousi2,3

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.


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 𝔟J(R), M is complete with respect to the 𝔟-adic topology, {Pi}i1 is a countable family of prime submodules of M, and xM, then x+Ni1Pi implies that x+NPj for some i1. This extends a theorem of Sharp and Vámos concerning prime ideals to prime submodules.