<abstract xmlns="http://www.w3.org/1999/xhtml">

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

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

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Notes on the zero-divisor graph and annihilating-ideal graph of a reduced ring"}

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...

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

Published Online: Jul 08, 2021

Page range: 51 - 70

Received: Aug 25, 2020

Accepted: Dec 30, 2020

DOI: https://doi.org/10.2478/auom-2021-0018

Keywords
Zero-divisor graph, Annihilating-ideal graph, Fixed-place ideal, Ring of real-valued continuous, Zariski-topology

© 2021 Mehdi Badie, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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

Published Online: Jul 08, 2021

Page range: 51 - 70

Received: Aug 25, 2020

Accepted: Dec 30, 2020

DOI: https://doi.org/10.2478/auom-2021-0018

KeywordsZero-divisor graph, Annihilating-ideal graph, Fixed-place ideal, Ring of real-valued continuous, Zariski-topology

© 2021 Mehdi Badie, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Zero-divisor graph, Annihilating-ideal graph, Fixed-place ideal, Ring of real-valued continuous, Zariski-topology