The inclusion prime ideal graph of a semigroup and ring

We introduce the inclusion prime ideal graph Inp(S) of nontrivial prime ideals of a commutative semigroup S. We characterize a semigroup S for which the graph Inp(S) is null, complete or connected. Then we study various graph parameters, thickness, metric and partition dimension of the inclusion prime ideal graph of the multiplicative semigroup Zn of integers of modulo n. Finally we characterize a ring R for which the graph Inp(R) is null, complete or connected and also some graph theoretic and ring theoretic properties are studied.