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Reduced rank in $\sigma[M]$ by Beachy, John A.; Medina-Bárcenas, Mauricio is a scholarly article available to read on EtoBox.

What is Reduced rank in $\sigma[M]$ about?

Using the concept of prime submodule introduced by Raggi et.al. we extend the notion of reduced rank to the module-theoretic context. 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. Later we 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 characterize when the module of quotients of $M$, respect to this torsion theory, has finite length in the quotient category. At the end we give conditions on a module $M$ with endomorphism ring $S$ to get that $S$ is an order in an Artininan ring, extending a remarkable Theorem of L.W. Small.

Author

Beachy, John A.; Medina-Bárcenas, Mauricio

Published

2022

Language

EN