Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On small dual rings   A ring $R$ is called right (small) dual if every (small) right ideal of $R$
is a right annihilator. Left (small) dual rings can be defined similarly. And a
ring $R$ is called (small) dual if $R$ is left and right (small) dual. It is
proved that $R$ is a dual ring if and only if $R$ is a semilocal and small dual
ring. Several known results are generalized and properties of small dual rings
are explored. As applications, some characterizations of QF rings are obtained
through small dualities of rings.

23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
17
The categories of this paper are: q-fin
16
The categories of this paper are: stat
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc