A note on almost prime submodule of CSM module over principal ideal domain

Abstract

An almost prime submodule is a generalization of prime submodule introduced in 2011 by Khashan. This algebraic structure was brought from an algebraic structure in ring theory, prime ideal, and almost prime ideal. This paper aims to construct similar properties of prime ideal and almost prime ideal from ring theory to module theory. The problem that we want to eliminate is the multiplication operation, which is missing in module theory. We use the definition of module annihilator to bridge the gap. This article gives some properties of the prime submodule and almost prime submodule of CMS module over a principal ideal domain. A CSM module is a module that every cyclic submodule. One of the results is that the idempotent submodule is an almost prime submodule.

The Characterization of Almost Prime Submodule on the 6 Finitely Generated Module over Principal Ideal Domain

“…A prime submodule by definition is an almost prime submodule, the converse is not always true. In some cases, we found that an almost prime submodule is a prime submodule, such as in a cyclic module over principal ideal domain or in a CSM module over a principal ideal domain [2]. Even in a free module over a principal ideal domain, when the rank of its submodule is less than its module, the almost prime submodule is a prime submodule [3].…”

Completely Prime Modules for Graded Simple Modules Which is Simple Over Leavitt Path Algebra

“…A proper prime submodule of if with , and implies or (Risnawita et al, 2021). Wardhana (Wardhana et al, 2021) and Saleh (Saleh et al, 2016) were characterized as prime submodules. A module (Risnawita et al, 2021 )is said to be a prime module over ring if with and implies that .…”

The Decomposition of a Finitely Generated Module over Some Special Ring

“…the CMS module characterization (Wardhana et al, 2021), U-complex module (Elfiyanti et al, 2020), F-CS-Rickart modules (Kaewwangsakoon & Pianskool, 2020), Injectivity & Projectivity of a module (Hijriati et al, 2018), uniserial module (Fitriani et al, 2021), linear codes (Irwansyah & Suprijanto, 2018), T-small modules (Sangwirotjanapat & Pianskool, 2018), the characteristics of a weakly prime submodule (Steven & Irawati, 2018), or the characterization of Bezout module (Ali Misri et al, 2013) and generalized Bezout module (Misri et al, 2016). In this study, we will generalize the decomposition, specifically from the principal ideal domain into the Dedekind domain.…”