On the Structure of Quotient Rings R / P via Identities with Multiplicative (Generalized) Derivations

Ali Yahya Hummdi, Hafedh Alnoghashi, Radwan M. Al-omary () and Rwaida A. Bahah
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Ali Yahya Hummdi: Department of Mathematics, College of Science, King Khalid University, Abha 61471, Saudi Arabia
Hafedh Alnoghashi: Department of Basic Sciences, College of Engineering and IT, Amran University, Amran, Yemen
Radwan M. Al-omary: Department of Mathematics, Ibb University, Ibb, Yemen
Rwaida A. Bahah: Department of Mathematics, Mahrah University, Mahrah, Yemen

Mathematics, 2025, vol. 13, issue 19, 1-12

Abstract: This work investigates the structure of an arbitrary ring R that contains a two-sided ideal I and a prime ideal P satisfying the condition P ⊊ I . Our analysis centers on the consequences of several identities that involve three multiplicative (generalized) derivations, denoted by Θ 1 , Θ 2 , Θ 3 : R → R . These are associated with maps θ 1 , θ 2 , θ 3 : R → R , which are not presumed to be additive or to be derivations themselves. The study further incorporates a non-zero derivation Δ along with two arbitrary, potentially non-additive, maps Γ 1 , Γ 2 : R → R . We establish conditions under which these identities lead to significant structural properties of the ring. To underscore the importance of our assumptions, we construct an example demonstrating that the primeness hypothesis on the ideal P is indispensable for our main conclusions.

Keywords: prime ideal; quotient ring; multiplicative (generalized) derivation; commutativity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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