### COMAXIMAL SUBMODULE GRAPHS OF UNITARY MODULES

#### Abstract

In this paper, a new kind of graph on a unitary module A over a commutative

ring R with identity, namely the co-maximal submodule graph is dened and studied as

a natural generalization of the comaximal ideal graph of a commutative ring R, denoted

by C(R). We use C(A) to denote this graph, with its vertices the proper submodules

of A which are not contained in the Jacobson radical of A, and two vertices B1 and B2

are adjacent if and only if B_1 + B_2 = A. We show some properties of this graph and

compare some of the results of C(A) and C(R). For example, this graph is a simple,

connected graph with diameter less than or equal to three, and both the clique number

and the chromatic number of the graph are equal to the number of maximal submodules

of the module A. It is shown that C(R) is isomorphic to a subgraph of C(A) when A is

a nitely generated cancellation (in particular, a nitely generated free) R-module. We

also discuss the conditions under which A is a nite direct sum of simple modules, C(A)

is isomorphic to a finite Boolean graph, and C(A) and C(R) are isomorphic graphs.

ring R with identity, namely the co-maximal submodule graph is dened and studied as

a natural generalization of the comaximal ideal graph of a commutative ring R, denoted

by C(R). We use C(A) to denote this graph, with its vertices the proper submodules

of A which are not contained in the Jacobson radical of A, and two vertices B1 and B2

are adjacent if and only if B_1 + B_2 = A. We show some properties of this graph and

compare some of the results of C(A) and C(R). For example, this graph is a simple,

connected graph with diameter less than or equal to three, and both the clique number

and the chromatic number of the graph are equal to the number of maximal submodules

of the module A. It is shown that C(R) is isomorphic to a subgraph of C(A) when A is

a nitely generated cancellation (in particular, a nitely generated free) R-module. We

also discuss the conditions under which A is a nite direct sum of simple modules, C(A)

is isomorphic to a finite Boolean graph, and C(A) and C(R) are isomorphic graphs.

#### Keywords

Co-maximal (submodule, ideal) graph; (nitely generated; cancellation; content [in particular, free, projective]) module; number of maximal submodules; injector ideal; connectedness and diameter; clique number; weakly perfect; Boolean graph

#### Full Text:

PDFDOI: http://dx.doi.org/10.14510%2Flm-ns.v37i1.1386