Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Integral degree of a ring and reduction numbers   The supremum of reduction numbers of ideals having principal reductions is
expressed in terms of the integral degree, a new invariant of the ring, which
is finite provided the ring has finite integral closure. As a consequence, one
obtains bounds for the Castelnuovo-Mumford regularity of the Rees algebra and
for the Artin-Rees numbers.

21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
19
The categories of this paper are: eess