Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On weakly S-prime submodules   Let $R$ be a commutative ring with a non-zero identity, $S$ be a
multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this
paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=\phi$ to be
weakly $S$-prime if there exists $s\in S$ such that whenever $a\in R$ and $m\in
M$ with $0\neq am\in N$, then either $sa\in(N:_{R}M)$ or $sm\in N$. Many
properties, examples and characterizations of weakly $S$-prime submodules are
introduced, especially in multiplication modules. Moreover, we investigate the
behavior of this structure under module homomorphisms, localizations, quotient
modules, cartesian product and idealizations. Finally, we define two kinds of
submodules of the amalgamation module along an ideal and investigate conditions
under which they are weakly $S$-prime.

15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
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The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
21
The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
11
The categories of this paper are: hep-lat
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: alg-geom