Kyungpook Mathematical Journal 2022; 62(3): 595-613
Published online September 30, 2022
Copyright © Kyungpook Mathematical Journal.
An Ideal-based Extended Zero-divisor Graph on Rings
Mohammad Ashraf, Mohit Kumar*
Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India
e-mail : mashraf80@hotmail.com
Department of Mathematics, Institute of Applied Sciences and Humanities, GLA University Mathura-281 406, Uttar Pradesh, India
e-mail : mohitkumaramu123@gmail.com
Received: June 10, 2019; Revised: April 13, 2020; Accepted: August 18, 2020
Let R be a commutative ring with identity and let I be a proper ideal of R. In this paper, we study the ideal based extended zero-divisor graph ΓI′(R) and prove that ΓI′(R) is connected with diameter at most two and if ΓI′(R) contains a cycle, then girth is at most four girth at most four. Furthermore, we study affinity the connection between the ideal based extended zero-divisor graph ΓI′(R) and the ideal-based zero-divisor graph ΓI(R) associated with the ideal I of R. Among the other things, for a radical ideal of a ring R, we show that the ideal-based extended zero-divisor graph ΓI′(R) is identical to the ideal-based zero-divisor graph ΓI(R) if and only if R has exactly two minimal prime-ideals which contain I.
Keywords: connected, diameter, girth, annihilator, prime ideal, Prime radical