Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Modules with ascending chain condition on annihilators and Goldie
  modules   Using the concepts of prime module, semiprime module and the concept of
ascending chain condition (ACC) on annihilators for an $R$-module $M$ . We
prove that if \ $M$ is semiprime \ and projective in $\sigma \left[ M\right] $,
such that $M$ satisfies ACC on annihilators, then $M$ has finitely many minimal
prime submodules. Moreover if each submodule $N\subseteq M$ contains a uniform
submodule, we prove that there is a bijective correspondence between a complete
set of representatives of isomorphism classes of indecomposable non
$M$-singular injective modules in $\sigma \left[ M\right] $ and the set of
minimal primes in $M$. If $M$ is Goldie module then $% \hat{M}\cong
E_{1}^{k_{1}}\oplus E_{2}^{k_{2}}\oplus ...\oplus E_{n}^{k_{n}}$ where each
$E_{i}$ is a uniform $M$-injective module. As an application, new
characterizations of left Goldie rings are obtained.

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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-alg
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The categories of this paper are: q-bio
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: plasm-ph