An Ideal-based Extended Zero-divisor Graph on Rings

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 Gamma ' I (R) and prove that Gamma ' I(R) is connected with diameter at most two and if Gamma ' I(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 Gamma ' I(R) and the ideal-based zero-divisor graph Gamma I(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 Gamma ' I (R) is identical to the ideal-based zero-divisor graph Gamma I (R) if and only if R has exactly two minimal prime-ideals which contain I.