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

[摘要] 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 dtt(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 dtt(AG(R)) is finite if and only if Min(R) is finite.