The annihilating-submodule graph of modules over commutative rings II

Abstract

Let M be a module over a commutative ring R. The annihilatingsubmodule graph of M , denoted by AG(M ), is a simple graph in which a non-zero submodule N of M is a vertex if and only if there exists a non-zero proper submodule K of M such that N K = (0), where N K, the product of N and K, is denoted by (N : M )(K : M )M and two distinct vertices N and K are adjacent if and only if N K = (0). This graph is a submodule version of the annihilating-ideal graph and under some conditions, is isomorphic with an induced subgraph of the Zariski topology-graph G(τ T ) which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283-3296). In this paper, we study the domination number of AG(M ) and some connections between the graph-theoretic properties of AG(M ) and algebraic properties of module M . 2010 Mathematics Subject Classification. 13C13, 13C99, 05C75.

The Zariski Topology-Graph of Modules Over Commutative Rings

“…SoM is not a prime module by [4,Remark 2.6] and a similar manner in proof of Theorem 2.11, shows thatM has a finite length so that R/Ann(M ) is an Artinian ring. As in the proof of part (a), M ∼ =M1 ⊕M 2 for some idempotent e ∈ R. IfM 1 has one non-trivial submodule N , then deg(Q 1 ⊕ M 2 ) > deg(N ⊕ M 2 ) (we note that by [6,Proposition 2.5],NK = (0) for some (0) =K <M 1 ) and this contradicts the regularity of G(τ T ). HenceM 1 is a simple module.…”

“…Remark 4. 6. Assume that S is a multiplicatively closed subset of R such that S ∩ (∪ P ∈T (P : M )) = ∅.…”

“…There are many papers on assigning graphs to rings or modules (see, for example, [1,5,6,9]). In [4], the present authors introduced and studied the graph G(τ T ) (resp.…”

The Zariski topology-graph of modules over commutative rings II

“…There are many papers on assigning graphs to rings or modules (see, for example, [1,5,6,9]). In [4], the present authors introduced and studied the graph G(τ T ) and AG(M), called the Zariski topology-graph and the annihilating-submodule graph, respectively.…”

On the graph of modules over commutative rings II

“…Then there exists an idempotent e in uR such that e − u ∈ I. (See[6, Lemma 2.4].) Let N be a minimal submodule of M and let Ann(M) be a nil ideal.…”