Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On some conditions on a Noetherian ring   In this paper for a noetherian ring R with nilradical N we define semiprime
ideals P and Q called as the left and right krull homogenous parts of N . We
also recall the known definitions of localisability and the weak ideal
invariance (w.i.i for short ) of an ideal of a noetherian ring R . We then
state and prove results that culminate in our main theorem whose statement is
given below ; Theorem :- Let R be a noetherian ring with nilradical N . Let P
and Q be semiprime ideals of R that are the right and left krull homogenous
parts of N respectively . Then the following conditions are equivalent ; (i) N
is a right w.i.i ideal of R ( respectively N is a left w.i.i ideal of R ) .
(ii)P is a right localizable ideal of R ( respectively Q is a left localizable
ideal of R ) .

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The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
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The categories of this paper are: hep-th
8
The categories of this paper are: hep-ex
15
The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
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The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th