Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Weakly left localizable rings   A new class of rings, {\em the class of weakly left localizable rings}, is
introduced. A ring $R$ is called {\em weakly left localizable} if each
non-nilpotent element of $R$ is invertible in some left localization $S^{-1}R$
of the ring $R$. Explicit criteria are given for a ring to be a weakly left
localizable ring provided the ring has only finitely many maximal left
denominator sets (eg, this is the case if a ring has a left Artinian left
quotient ring). It is proved that a ring with finitely many maximal left
denominator sets that satisfies some natural conditions is a weakly left
localizable ring iff its left quotient ring is a direct product of finitely
many local rings such that their radicals are nil ideals.

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The categories of this paper are: nucl-th
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: cs
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: mtrl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: astro-ph