International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 58132, 8 pages
Research Article

δ-Small Submodules and δ-Supplemented Modules

Yongduo Wang

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

Received 9 January 2007; Accepted 18 June 2007

Academic Editor: Akbar Rhemtulla

Copyright © 2007 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 right R-module. It is shown that (1) δ(M) is Noetherian if and only if M satisfies ACC on δ-small submodules; (2) δ(M) is Artinian if and only if M satisfies DCC on δ-small submodules; (3) M is Artinian if and only if M is an amply δ-supplemented module and satisfies DCC on δ-supplement submodules and on δ-small submodules.