International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 69, Pages 4373-4387
doi:10.1155/S016117120320346X
On some properties of ⊕-supplemented modules
Département de Mathématiques, Faculté des Sciences de Tétouan, Université Abdelmalek Essaâdi, B.P 21.21, Tétouan, Morocco
Received 15 March 2002
Copyright © 2003 A. Idelhadj and R. Tribak. 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
A module M is ⊕-supplemented if every submodule of M has a supplement which is a direct summand of M. In this paper, we show that a quotient of a ⊕-supplemented module is not in general ⊕-supplemented. We prove that over a commutative ring R, every finitely generated ⊕-supplemented R-module M having dual Goldie dimension less than or equal to three is a direct sum of local modules. It is also shown that a ring R is semisimple if and only if the class of ⊕-supplemented R-modules coincides with the class of injective R-modules. The structure of ⊕-supplemented modules over a commutative principal ideal ring is completely determined.