Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Reduced rank in $\sigma[M]$   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 $S$ in order to get that $S$
is an order in an Artininan ring, extending Small's Theorem.

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The categories of this paper are: cs
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: math-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: supr-con