International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 9, Pages 1455-1460

Generalizations of Hopfian and co-Hopfian modules

Yongduo Wang

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China

Received 25 September 2004

Copyright © 2005 Yongduo Wang. 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 R be a ring and M a left R-module. M which satisfies DCC on essential submodules is GCH, and M which satisfies ACC on small submodules is WH. If M[X] is GCH R[X]-module, then M is GCH R-module. Examples show that a GCH module need not be co-Hopfian and a WH module need not be Hopfian.