Shokoufeh Habibi, Zohreh Habibi, Masoomeh Hezarjaribi: Domination number in the Zariski topology-graph of modules, 329-340

Abstract:

Let $M$ be a module over a commutative ring and let $Spec(M)$ be the collection of all prime submodules of $M$. One can define a Zariski topology on $Spec(M)$, which is analogous to that on $Spec(R)$, and then for any non-empty set $T$ of $Spec(M)$, it is possible to define a simple graph $G(\tau_T)$, called the Zariski topology-graph. In this paper, we study the domination number of $G(\tau_T)$ and some connections between the graph-theoretic properties of $G(\tau_T)$ and algebraic properties of the module $M$.

Key Words: Rings and modules, Zariski topology, graph, domination number.

2010 Mathematics Subject Classification: Primary 13C13, 13C99; Secondary 05C75.