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Reduced rank in $σ[M]$

John A. Beachy, Mauricio Medina-Bárcenas

TL;DR

The paper extends the notion of reduced rank to the module-theoretic setting in $\sigma[M]$ by employing prime submodules and the prime radical $\mathfrak{L}(M)$. It develops a localization framework using torsion theories $\gamma$ and $\tau_g$, showing that the quotient category $\sigma[M]/\tau_g$ is spectral and often equivalent to $\mathrm{Mod}$-$T$ with $T=\mathrm{End}_R(\widehat{M})$, while $\mathcal{Q}_{\tau_g}(M)$ is semisimple of finite length. Finite reduced rank is characterized by several equivalent conditions (e.g., $M/\mathfrak{L}(M)$ Goldie and $\mathfrak{L}(M)^k\subseteq \gamma(M)$) and is shown to be Morita invariant, with stability under submodules, sums, and corner constructions. A generalization of Small's Theorem links endomorphism rings $S=\mathrm{End}_R(M)$ and $T=\mathrm{End}_R(\mathcal{Q}_\gamma(M))$, showing $S$ is a right order in $T$ when finite reduced rank holds and deriving consequences for Rickart modules and semicentral idempotents. Overall, the work unifies reduced rank, Goldie theory, and endomorphism-ring perspectives in a broad module-theoretic context, providing practical criteria for when corner and matrix rings preserve finite reduced rank.

Abstract

Using the concept of prime submodule introduced by Raggi et.al. we extend the notion of reduced rank to the module-theoretic context of $σ[M]$. We study the quotient category of $σ[M]$ modulo the hereditary torsion theory cogenerated by the $M$-injective hull of $M$, when $M$ is a semiprime Goldie module. We prove that this quotient category is spectral. We then consider the hereditary torsion theory in $σ[M]$ cogenerated by the $M$-injective hull of $M/\mathfrak{L}(M)$, where $\mathfrak{L}(M)$ is the prime radical of $M$, and we determine when the module of quotients of $M$, with respect to this torsion theory, has finite length in the quotient category. Finally, we give conditions on a module $M$ with endomorphism ring $S$ under which $S$ is an order in an Artinian ring, extending Small's Theorem.

Reduced rank in $σ[M]$

TL;DR

The paper extends the notion of reduced rank to the module-theoretic setting in by employing prime submodules and the prime radical . It develops a localization framework using torsion theories and , showing that the quotient category is spectral and often equivalent to - with , while is semisimple of finite length. Finite reduced rank is characterized by several equivalent conditions (e.g., Goldie and ) and is shown to be Morita invariant, with stability under submodules, sums, and corner constructions. A generalization of Small's Theorem links endomorphism rings and , showing is a right order in when finite reduced rank holds and deriving consequences for Rickart modules and semicentral idempotents. Overall, the work unifies reduced rank, Goldie theory, and endomorphism-ring perspectives in a broad module-theoretic context, providing practical criteria for when corner and matrix rings preserve finite reduced rank.

Abstract

Using the concept of prime submodule introduced by Raggi et.al. we extend the notion of reduced rank to the module-theoretic context of . We study the quotient category of modulo the hereditary torsion theory cogenerated by the -injective hull of , when is a semiprime Goldie module. We prove that this quotient category is spectral. We then consider the hereditary torsion theory in cogenerated by the -injective hull of , where is the prime radical of , and we determine when the module of quotients of , with respect to this torsion theory, has finite length in the quotient category. Finally, we give conditions on a module with endomorphism ring under which is an order in an Artinian ring, extending Small's Theorem.
Paper Structure (5 sections, 39 theorems, 26 equations)

This paper contains 5 sections, 39 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Hereditary Torsion Theories in $\sigma [M]$
  3. The quotient category of $\sigma [M]$ with $M$ a semiprime Goldie module
  4. Modules with finite reduced rank in $\sigma[M]$
  5. A generalization of Small's Theorem

Key Result

Lemma 2.1

Let $M$ be an $R$-module. If $\tau=(\mathcal{T},\mathcal{F})$ is a pair of classes of modules in $\sigma[M]$, then $\tau$ is a hereditary torsion theory in $\sigma[M]$ if and only if there exists a hereditary torsion theory $(\mathfrak{T},\mathfrak{F})$ in $R$-Mod such that $\mathcal{T}=\mathfrak{T}

Theorems & Definitions (85)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • ...and 75 more