On Prime Ideals in Rings Satisfying Certain Functional Identities
In this note, we will examine the relationship between a generalized derivation <i>ϝ</i> associated with a derivation <i>∂</i> that satisfies specific algebraic identities and the behavior of a factor ring <inline-formula><math xmlns="http://www.w3.org/1998/Math...
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MDPI AG
2025-03-01
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| Online Access: | https://www.mdpi.com/2075-1680/14/4/247 |
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| author | Kholood Alnefaie Radwan M. Al-omary |
| author_facet | Kholood Alnefaie Radwan M. Al-omary |
| author_sort | Kholood Alnefaie |
| collection | DOAJ |
| description | In this note, we will examine the relationship between a generalized derivation <i>ϝ</i> associated with a derivation <i>∂</i> that satisfies specific algebraic identities and the behavior of a factor ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>/</mo><mi>P</mi></mrow></semantics></math></inline-formula>, where <i>P</i> is a prime ideal of any ring <i>S</i>. We will also explore several consequences and special cases as applications of our findings. Additionally, various examples have been provided to highlight the necessity of the primeness hypothesis in our theorems. |
| format | Article |
| id | doaj-art-9b58bf09bdeb43008ec45db5d0492fe3 |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-9b58bf09bdeb43008ec45db5d0492fe32025-08-20T02:24:39ZengMDPI AGAxioms2075-16802025-03-0114424710.3390/axioms14040247On Prime Ideals in Rings Satisfying Certain Functional IdentitiesKholood Alnefaie0Radwan M. Al-omary1Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi ArabiaDepartment of Mathematics, Ibb University, Ibb 70270, YemenIn this note, we will examine the relationship between a generalized derivation <i>ϝ</i> associated with a derivation <i>∂</i> that satisfies specific algebraic identities and the behavior of a factor ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>/</mo><mi>P</mi></mrow></semantics></math></inline-formula>, where <i>P</i> is a prime ideal of any ring <i>S</i>. We will also explore several consequences and special cases as applications of our findings. Additionally, various examples have been provided to highlight the necessity of the primeness hypothesis in our theorems.https://www.mdpi.com/2075-1680/14/4/247prime idealfactor ringintegral domaingeneralized derivations |
| spellingShingle | Kholood Alnefaie Radwan M. Al-omary On Prime Ideals in Rings Satisfying Certain Functional Identities Axioms prime ideal factor ring integral domain generalized derivations |
| title | On Prime Ideals in Rings Satisfying Certain Functional Identities |
| title_full | On Prime Ideals in Rings Satisfying Certain Functional Identities |
| title_fullStr | On Prime Ideals in Rings Satisfying Certain Functional Identities |
| title_full_unstemmed | On Prime Ideals in Rings Satisfying Certain Functional Identities |
| title_short | On Prime Ideals in Rings Satisfying Certain Functional Identities |
| title_sort | on prime ideals in rings satisfying certain functional identities |
| topic | prime ideal factor ring integral domain generalized derivations |
| url | https://www.mdpi.com/2075-1680/14/4/247 |
| work_keys_str_mv | AT kholoodalnefaie onprimeidealsinringssatisfyingcertainfunctionalidentities AT radwanmalomary onprimeidealsinringssatisfyingcertainfunctionalidentities |