Notes on the zero-divisor graph and annihilating-ideal graph of a reduced ring

Abstract

We translate some graph properties of 𝔸𝔾(R) and Γ(R) to some topological properties of Zariski topology. We prove that the facts “(1) The zero ideal of R is an anti fixed-place ideal. (2) Min(R) does not have any isolated point. (3) Rad(𝔸𝔾 (R)) = 3. (4) Rad(Γ(R)) = 3. (5) Γ(R) is triangulated (6) 𝔸𝔾 (R) is triangulated.” are equivalent. Also, we show that if the zero ideal of a ring R is a fixed-place ideal, then dt t (𝔸𝔾 (R)) = |ℬ(R)| and also if in addition |Min(R)| > 2, then dt(𝔸𝔾 (R)) = |ℬ (R)|. Finally, it is shown that dt(𝔸𝔾 (R)) is finite if and only if dt t (𝔸𝔾 (R)) is finite if and only if Min(R) is finite.