Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The annihilating-submodule graph of modules over commutative rings II   Let M be a module over a commutative ring R. In this paper, we continue our
study of annihilating-submodule graph AG(M) which was introduced in (The
Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42
(2014), 3283{3296). AG(M) is a (undirected) graph in which a nonzero submodule
N of M is a vertex if and only if there exists a nonzero proper submodule K of
M such that NK = (0), where NK, the product of N and K, is defined by (N : M)(K
: M)M and two distinct vertices N and K are adjacent if and only if NK = (0).
We prove that if AG(M) is a tree, then either AG(M) is a star graph or a path
of order 4 and in the latter case M\cong F\times ?S, where F is a simple module
and S is a module with a unique non-trivial submodule. Moreover, we prove that
if M is a cyclic module with at least three minimal prime submodules, then
gr(AG(M)) = 3 and for every cyclic module M, cl(AG(M))\geq |Min(M)|.

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The categories of this paper are: math-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: supr-con
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The categories of this paper are: q-alg
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th