Some results on principal ideal graph of a ring

Abstract

Let R be an associative (not necessarily commutative) ring. In the present paper, we introduced a new type of graph (called ‘Principal Ideal Graph’, denoted by PIG(R)) related to a given associative ring R. We presented some examples. We obtained few fundamental important relations between rings and graphs with respect to the properties: simple ring, complete graph, etc. We also observed that if R and S are isomorphic rings, then the related principal ideal graphs are isomorphic, but the converse is not true. We defined an equivalence relation on a given ring R and obtained a one-to-one correspondence between the set of all equivalence classes and the set of all connected components of PIG(R). We introduced the concept ‘full Hamiltonian decomposition’ for a general graph, and proved that there exists a full Hamiltonian decomposition for PIG(R). Key words: Principal ideal graph, ring, Hamiltonian decomposition, complete graph.