# On Supplement Submodules

### Abstract

We investigate some properties of supplement submodules. Some relations between lying-above and weak supplement submodules are also studied. Let*V*be a supplement of a submodule

*U*in

*M*. Then it is possible to define a bijective map between the maximal submodules of

*V*and the maximal submodules of

*M*that contain

*U*. Let

*M*be an

*R*-module,

*U*≤

*M*, let

*V*be a weak supplement of

*U*, and let

*K*≤

*V*. In this case,

*K*is a weak supplement of

*U*if and only if

*V*lies above

*K*in

*M*. We prove that an

*R*-module

*M*is amply supplemented if and only if every submodule of

*M*lies above a supplement in

*M*. We also prove that

*M*is semisimple if and only if every submodule of

*M*is a supplement in

*M*.

Published

25.07.2013

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 65, no. 7, July 2013, pp. 961–966, https://umj.imath.kiev.ua/index.php/umj/article/view/2481.

Issue

Section

Research articles