<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let R be a Noetherian integral domain which is also an algebra over ℚ (ℚ is the field of rational numbers). Let σ be an endo-morphism of R and δ a σ-derivation of R. We recall that a ring R is a weak (σ, δ)-rigid ring if a(σ(a)+ δ(a)) <italic>∈</italic> N(R) if and only if a <italic>∈</italic> N(R) for a <italic>∈</italic> R (N(R) is the set of nilpotent elements of R). With this we prove that if R is a Noetherian integral domain which is also an algebra over ℚ, σ an automorphism of R and δ a σ-derivation of R such that R is a weak (σ, δ)-rigid ring, then N(R) is completely semiprime.</p></abstract>

Let R be a Noetherian integral domain which is also an algebra over ℚ (ℚ is the field of rational numbers). Let σ be an endo-morphism of R and δ a σ-derivation of R. We recall that a ring R is a weak (σ, δ)-rigid ring if a(σ(a)+ δ(a)) ∈ N(R) if and only if a ∈ N(R) for a ∈ R (N(R) is the set of nilpotent elements of R). With this we prove that if R is a Noetherian integral domain which is also an algebra over ℚ, σ an automorphism of R and δ a σ-derivation of R such that R is a weak (σ, δ)-rigid ring, then N(R) is completely semiprime.

On weak (σ, δ)-rigid rings over Noetherian rings

Acta Universitatis Sapientiae, Mathematica

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

{"article-title":"On weak (σ, δ)-rigid rings over Noetherian rings"}

Let R be a Noetherian integral domain which is also an algebra over ℚ (ℚ is the field of rational numbers). Let σ be an endo-morphism of R and δ a σ-derivation...

On weak (σ, δ)-rigid rings over Noetherian rings

Published Online: Jul 16, 2020

Page range: 5 - 13

Received: Dec 16, 2019

DOI: https://doi.org/10.2478/ausm-2020-0001

Keywords
automorphism, derivation, nilpotent element, completely semiprime ideal

© 2020 Vijay Kumar Bhat et al., published by Sciendo

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

On weak (σ, δ)-rigid rings over Noetherian rings

Published Online: Jul 16, 2020

Page range: 5 - 13

Received: Dec 16, 2019

DOI: https://doi.org/10.2478/ausm-2020-0001

Keywordsautomorphism, derivation, nilpotent element, completely semiprime ideal

© 2020 Vijay Kumar Bhat et al., published by Sciendo

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

Keywords
automorphism, derivation, nilpotent element, completely semiprime ideal