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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2024-12-01
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| author | Ali Yahya Hummdi Zakia Z. Al-Amery Radwan M. Al-omary |
| author_facet | Ali Yahya Hummdi Zakia Z. Al-Amery Radwan M. Al-omary |
| author_sort | Ali Yahya Hummdi |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-037d26985c564950a32ad6727d8afbdb |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-037d26985c564950a32ad6727d8afbdb2025-01-24T13:22:09ZengMDPI AGAxioms2075-16802024-12-011411510.3390/axioms14010015Factor Rings with Algebraic Identities via Generalized DerivationsAli Yahya Hummdi0Zakia Z. Al-Amery1Radwan M. Al-omary2Department of Mathematics, College of Science, King Khalid University, Abha 61471, Saudi ArabiaDepartment of Mathematics, Aden University, Aden 5243, YemenDepartment of Mathematics, Ibb University, Ibb 70270, YemenThe 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.https://www.mdpi.com/2075-1680/14/1/15prime idealfactor ringintegral domaingeneralized derivations |
| spellingShingle | Ali Yahya Hummdi Zakia Z. Al-Amery Radwan M. Al-omary Factor Rings with Algebraic Identities via Generalized Derivations Axioms prime ideal factor ring integral domain generalized derivations |
| title | Factor Rings with Algebraic Identities via Generalized Derivations |
| title_full | Factor Rings with Algebraic Identities via Generalized Derivations |
| title_fullStr | Factor Rings with Algebraic Identities via Generalized Derivations |
| title_full_unstemmed | Factor Rings with Algebraic Identities via Generalized Derivations |
| title_short | Factor Rings with Algebraic Identities via Generalized Derivations |
| title_sort | factor rings with algebraic identities via generalized derivations |
| topic | prime ideal factor ring integral domain generalized derivations |
| url | https://www.mdpi.com/2075-1680/14/1/15 |
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