Reduced rank in $\sigma[M]$

TL;DR

This paper extends the concept of reduced rank to modules in the category [M], analyzing quotient categories, torsion theories, and conditions for endomorphism rings to be orders in Artinian rings, with implications for module theory.

Contribution

It introduces a module-theoretic notion of reduced rank in [M] and explores its properties, including spectral quotient categories and conditions for endomorphism rings to be Artinian orders.

Findings

The quotient category of [M] modulo a certain hereditary torsion theory is spectral.

Conditions are established for the module of quotients to have finite length in the quotient category.

Criteria are provided for the endomorphism ring of a module to be an order in an Artinian ring.

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 $\sigma[M]$. We study the quotient category of $\sigma[M]$ 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 $\sigma[M]$ 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.