Classification of multiplication modules over multiplication rings with finitely many minimal primes

TL;DR

The paper addresses the classification of multiplication modules over multiplication rings with finitely many minimal primes. It shows that the ring decomposes as $R \cong \prod_{i=1}^{n} D_i$, and that a module $M=\bigoplus_{i=1}^{n} M_i$ is a multiplication module precisely when each $M_i$ is isomorphic to $D_i$ or to a quotient $D_i/I_i$ (or to a nonzero ideal in the Dedekind case), with detailed treatment for Dedekind domains and Artinian local principal ideal rings. It proves that a faithful, Noetherian, distributive $R$-module is equivalent to a faithful multiplication $R$-module, and it describes the endomorphism ring as a multiplication ring via $\operatorname{End}_R(M) \simeq R/\operatorname{ann}_R(M)$. The results provide a constructive, structure-driven classification linking the ring’s finite minimal prime structure to module-distributivity and localization, with implications for endomorphism behavior and module decomposition.

Abstract

A classification of multiplication modules over multiplication rings with finitely many minimal primes is obtained. A characterisation of multiplication rings with finitely many minimal primes is given via faithful, Noetherian, distributive modules. It is proven that for a multiplication ring with finitely many minimal primes every faithful, Noetherian, distributive module is a faithful multiplication module, and vice versa.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Corollary 2.5