TY - JOUR AU - C. Nebiyev AU - A. Pancar PY - 2013/07/25 Y2 - 2024/03/28 TI - On Supplement Submodules JF - Ukrains’kyi Matematychnyi Zhurnal JA - Ukr. Mat. Zhurn. VL - 65 IS - 7 SE - Research articles DO - UR - https://umj.imath.kiev.ua/index.php/umj/article/view/2481 AB - 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. ER -