Survey on second submodules of modules over commutative rings

TL;DR

This survey consolidates the theory of second submodules in modules over commutative rings, presenting core definitions, dualities with prime submodules, and a spectrum $Spec^s(M)$. It systematically organizes a range of generalizations—$S$-second, $I$-second, graded, fuzzy, and $\psi$-second submodules—and examines their fundamental properties, interactions with localization, and annihilator structures. The compilation includes key finiteness and structural results in specialized module classes, notably coreduced comultiplication and graded contexts, providing a comprehensive reference for researchers. The work highlights the versatility and reach of second submodules across algebraic settings and suggests avenues for applications in lattice theory and related fields.

Abstract

Let R be a commutative ring with identity. The concept of second submodule of an R-module (as a dual notion of prime submodules) was introduced and studied by S.Yassemi in 2001. This notion has obtained a great attention by many authors and now there is a considerable amount of research concerning this class of modules. The main purpose of this paper is to collect these results and provide a useful source for those who are interested in research in this field.

Theorems & Definitions (267)

• Lemma 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Proposition 2.5
• Proposition 2.6
• Theorem 2.7
• Corollary 2.8
• Proposition 2.9
• Theorem 2.10