A little more on the zero-divisor graph and the annihilating-ideal graph of a reduced ring

TL;DR

This paper explores the relationship between graph properties of zero-divisor and annihilating-ideal graphs of reduced rings and their topological properties, establishing conditions for triangulation and finiteness related to the ring's minimal prime ideals.

Contribution

It provides new characterizations linking graph properties of AG(R) and Gamma(R) to topological features of the ring's spectrum, especially regarding triangulation and finiteness conditions.

Findings

Rad(Gamma(R)) and Rad(AG(R)) are equal and equal to 3 under certain conditions.

Gamma(R) and AG(R) are triangulated if and only if the zero ideal is an anti fixed-place ideal.

Finite dtt(AG(R)) and dt(AG(R)) correspond to the finiteness of Min(R).

Abstract

We have tried to translate some graph properties of AG(R) and Gamma(R) to the topological properties of Zariski topology. We prove that Rad(Gamma(R)) and Rad(AG(R)) are equal and they are equal to 3, if and only if the zero ideal of R is an anti fixed-place ideal, if and only if Min(R) does not have any isolated point, if and only if Gamma(R) is triangulated, if and only if AG(R) is triangulated. Also, we show that if the zero ideal of a ring R is a fixed-place ideal, then dtt(AG(R)) = |B(R)| and also if in addition |Min(R)| > 2, then dt(AG(R)) = |B(R)|. Finally, it has been shown that dt(AG(R)) is finite, if and only if dtt(AG(R) is finite; if and only if Min(R) is finite.