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For commutative rings with identity, we introduce and study the concept of semi <italic>r</italic>-ideals which is a kind of generalization of both <italic>r</italic>-ideals and semiprime ideals. A proper ideal <italic>I</italic> of a commutative ring <italic>R</italic> is called semi <italic>r</italic>-ideal if whenever <italic>a</italic>2 ∈ <italic>I</italic> and <italic>Ann</italic>R(<italic>a</italic>) = 0, then <italic>a</italic> ∈ <italic>I</italic>. Several properties and characterizations of this class of ideals are determined. In particular, we investigate semi <italic>r</italic>-ideal under various contexts of constructions such as direct products, localizations, homomorphic images, idealizations and amalagamations rings. We extend semi <italic>r</italic>-ideals of rings to semi <italic>r</italic>-submodules of modules and clarify some of their properties. Moreover, we define submodules satisfying the <italic>D</italic>-annihilator condition and justify when they are semi <italic>r</italic>-submodules.
</abstract>

For commutative rings with identity, we introduce and study the concept of semi r-ideals which is a kind of generalization of both r-ideals and semiprime ideals. A proper ideal I of a commutative ring R is called semi r-ideal if whenever a2 ∈ I and AnnR(a) = 0, then a ∈ I. Several properties and characterizations of this class of ideals are determined. In particular, we investigate semi r-ideal under various contexts of constructions such as direct products, localizations, homomorphic images, idealizations and amalagamations rings. We extend semi r-ideals of rings to semi r-submodules of modules and clarify some of their properties. Moreover, we define submodules satisfying the D-annihilator condition and justify when they are semi r-submodules.

Semi <italic>r</italic>-ideals of commutative rings

Department of Mathematics, Faculty of Science

Department of Basic Sciences, Faculty of Engineering

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":{"italic":"r","content":["Semi","-ideals of commutative rings"]}}

For commutative rings with identity, we introduce and study the concept of semi r-ideals which is a kind of generalization of both r-ideals and semiprime...

Semi r-ideals of commutative rings

Published Online: Mar 27, 2023

Page range: 101 - 126

Received: Jul 22, 2022

Accepted: Dec 29, 2022

DOI: https://doi.org/10.2478/auom-2023-0022

Keywords
Semiprime ideal, semiprime submodule, semi -ideal, semi -submodule

© 2023 Hani A. Khashan et al., published by Sciendo

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

Semi r-ideals of commutative rings

Published Online: Mar 27, 2023

Page range: 101 - 126

Received: Jul 22, 2022

Accepted: Dec 29, 2022

DOI: https://doi.org/10.2478/auom-2023-0022

KeywordsSemiprime ideal, semiprime submodule, semi -ideal, semi -submodule

© 2023 Hani A. Khashan et al., published by Sciendo

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

Keywords
Semiprime ideal, semiprime submodule, semi -ideal, semi -submodule