Pseudo-Primary, Classical Prime and Pseudo-Classical Primary Elements in Lattice Modules

TL;DR

This paper introduces and characterizes pseudo-primary and pseudo-classical primary elements in lattice modules, showing their radicals are prime and exploring properties of classical prime elements.

Contribution

It defines new types of elements in lattice modules and establishes their properties and relationships with prime radicals, advancing the theory of lattice modules.

Findings

Radicals of pseudo-primary elements are prime.

Characterizations of pseudo-primary and pseudo-classical primary elements.

Properties and characterizations of classical prime elements.

Abstract

In this paper, we introduce the notion of pseudo-primary elements and pseudo-classical primary elements in an $L$-module $M$ and obtain their characterizations. The aim of the paper is to show $rad(N)\in M$, the radical of $N\in M$ is prime if $N$ is either pseudo-primary or pseudo-classical primary. Also, we study classical prime elements of an $L$-module $M$ to obtain many of its characterizations and its properties.