Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A generalization of purely extending modules relative to a torsion
  theory   In this work we introduce a new concept, namely, $\tau_{s}$-extending modules
(rings) which is torsion-theoretic analogues of extending modules and then we
extend many results from extending modules to this new concept. For instance we
show that for any ring $R$ with unit, if $R$ is purely $\tau_{s}$-extending
then every cyclic $\tau$-nonsingular $R$-module is flat and we show that this
fact is true over a principal ideal domain as well. Also, we make a
classification for the direct sums of the rings to be purely
$\tau_{s}$-extending.

30
The categories of this paper are: q-alg
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph