Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Note On Noeherian Rings   In this paper we introduce the definition of a noetherian disjoint ring and
that of a noetherian non-disjoint ring . For a noetherian ring R , with
nilradical N if P and Q represent the semiprime ideals of R called as the right
and the left krull-homogenous parts of N as defined in [8] , then we prove the
main theorem of this paper for the ring R whose statement is given below. Main
Theorem :- Let R be a Noetherian ring with nilradical N . Let P and Q represent
the right and the left krull-homogenous parts of N . Then the following hold
true for the ring R ; (a) If R is a disjoint ring , then the nilradical N of R
is a right and a left weakly ideal invariant ideal of R . Hence N is a right
and a left localizable semiprime ideal of R . (b) If R is a non-disjoint ring
then the following are equivalent conditions on R ;
  (i) N is a right and a left weakly ideal invariant ideal of R .
  (ii) P = Q is a right and a left localizable semiprime ideal of R .

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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int