An extension of annihilating-ideal graph of commutative rings

Commun. Korean Math. Soc. Published online September 7, 2020

Mahtab Koohi Kerahroodi and Fatemeh Nabaei
Department of Mathematics, Malayer University, Malayer, Iran., Department of Mathematics, Malayer Branch, Islamic Azad University, Malayer, Iran.

Abstract : Let R be a commutative ring with unity. The extension of
annihilating-ideal graph of R is the graph whose vertices
are nonzero annihilating ideals of R and two distinct
vertices I and J are adjacent if and only if there exist
n,m ∈ N such that I^{n}J^{m}=(0) with I^{n}, J^{m} ̸= (0).
First, we differentiate when annihilating-ideal graph and its
extension coincide. Then, we have characterized the diameter
and the girth of extension of annihilating-ideal graph of R when
R is a finite direct products of rings. Moreover, we show that
extension of annihilating-ideal graph of R contains a cycle, if
it don't equal with annihilating-ideal graph of R.