journal-of-discrete-mathematical-sciences-and-cryptography Journal of Discrete Mathematical Sciences and Cryptography 2169-0065 0972-0529 Taru Publications 10.47974/JDMSC-2273 Pu–supplement sub-modules Hamdi Halah Qahtan Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq Al-Mothafar Nuhad S. Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq 28 4-B 1343 1353 26 06 2025

In this article, we present a notion such as purely supplement submodule,  also consider some properties of this concept, and research some relationships between purely supplement submodules and some other related ideas. Furthermore, we give amply purely supplemented modules and weakly purely supplemented modules.

 Supplement submodules Pure submodules Purely-supplement submodules Amply purely supplement submodules Weakly purely supplement submodules Purely-supplemented modules Amply purely supplemented modules Weakly purely supplemented modules access open retracted no