ON COMPLETELY PRIME SUBMODULES

Abstract

The formal study of completely prime modules was initiated by N. J. Groenewald and the current author in the paper: Completely prime submodules, Int. Electron. J. Algebra, 13(2013), 1–14. In this paper, the study of completely prime modules is continued. Firstly, the advantage completely prime modules have over prime modules is highlighted and different situations that lead to completely prime modules given. Later, emphasis is put on fully completely prime modules (i.e., modules whose all submodules are completely prime). For a fully completely prime left R-module M, if a, b ∈ R and m ∈ M, then abm = bam, am = a km for all positive integers k, and either am = abm or bm = abm. In the last section, two different torsion theories induced by the completely prime radical are given.

Keywords

• Domain, prime module, completely prime module, completely prime radical, torsion theory