Factor Rings with Algebraic Identities via Generalized Derivations

Factor Rings with Algebraic Identities via Generalized Derivations

The current article focuses on studying the behavior of a ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℜ</mo><mo>/</mo><mo>Π</mo></mrow></semantics&...

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Bibliographic Details
Main Authors: Ali Yahya Hummdi, Zakia Z. Al-Amery, Radwan M. Al-omary
Format: Article
Language:English
Published: MDPI AG 2024-12-01
Series:Axioms
Subjects:
prime ideal
factor ring
integral domain
generalized derivations
Online Access:https://www.mdpi.com/2075-1680/14/1/15
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Summary:The current article focuses on studying the behavior of a ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℜ</mo><mo>/</mo><mo>Π</mo></mrow></semantics></math></inline-formula> when <i>ℜ</i> admits generalized derivations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ψ</mo></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> with associated derivations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>, respectively. These derivations satisfy specific differential identities involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Π</mo></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Π</mo></semantics></math></inline-formula> is a prime ideal of an arbitrary ring <i>ℜ</i>, not necessarily prime or semiprime. Furthermore, we explore some consequences of our findings. To emphasize the necessity of the primeness of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Π</mo></semantics></math></inline-formula> in the hypotheses of our various theorems, we provide a list of examples.
ISSN:2075-1680

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