Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications
A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation are central, then the ring is commutative. In the present paper, we extended this bluestocking theorem to an arbitrary ring with involution involving...
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MDPI AG
2023-07-01
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| author | Shakir Ali Turki M. Alsuraiheed Mohammad Salahuddin Khan Cihat Abdioglu Mohammed Ayedh Naira N. Rafiquee |
| author_facet | Shakir Ali Turki M. Alsuraiheed Mohammad Salahuddin Khan Cihat Abdioglu Mohammed Ayedh Naira N. Rafiquee |
| author_sort | Shakir Ali |
| collection | DOAJ |
| description | A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation are central, then the ring is commutative. In the present paper, we extended this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we established the ∗-version of Vukman’s theorem. Precisely, we describe the structure of quotient ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">A</mi><mo>/</mo><mi mathvariant="fraktur">L</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">A</mi></semantics></math></inline-formula> is an arbitrary ring and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">L</mi></semantics></math></inline-formula> is a prime ideal of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">A</mi></semantics></math></inline-formula>. Further, by taking advantage of the ∗-version of Vukman’s theorem, we show that if a 2-torsion free semiprime <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">A</mi></semantics></math></inline-formula> with involution admits a nonzero ∗-centralizing derivation, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">A</mi></semantics></math></inline-formula> contains a nonzero central ideal. This result is in the spirit of the classical result due to Bell and Martindale (Theorem 3). As the applications, we extended and unified several classical theorems. Finally, we conclude our paper with a direction for further research. |
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| spelling | doaj.art-ac9258a97f2240b1b5661d0ff14cfa102023-11-18T20:20:48ZengMDPI AGMathematics2227-73902023-07-011114311710.3390/math11143117Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with ApplicationsShakir Ali0Turki M. Alsuraiheed1Mohammad Salahuddin Khan2Cihat Abdioglu3Mohammed Ayedh4Naira N. Rafiquee5Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, IndiaDepartment of Mathematics, King Saud University, Riyadh 11495, Saudi ArabiaDepartment of Applied Mathematics, Z. H. College of Engineering & Technology, Aligarh Muslim University, Aligarh 202002, IndiaDepartment of Mathematics & Science Education, Karamanoglu Mehmetbey University, Karaman 70100, TurkeyDepartment of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, IndiaDepartment of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, IndiaA well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation are central, then the ring is commutative. In the present paper, we extended this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we established the ∗-version of Vukman’s theorem. Precisely, we describe the structure of quotient ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">A</mi><mo>/</mo><mi mathvariant="fraktur">L</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">A</mi></semantics></math></inline-formula> is an arbitrary ring and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">L</mi></semantics></math></inline-formula> is a prime ideal of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">A</mi></semantics></math></inline-formula>. Further, by taking advantage of the ∗-version of Vukman’s theorem, we show that if a 2-torsion free semiprime <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">A</mi></semantics></math></inline-formula> with involution admits a nonzero ∗-centralizing derivation, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">A</mi></semantics></math></inline-formula> contains a nonzero central ideal. This result is in the spirit of the classical result due to Bell and Martindale (Theorem 3). As the applications, we extended and unified several classical theorems. Finally, we conclude our paper with a direction for further research.https://www.mdpi.com/2227-7390/11/14/3117derivation∗-centralizing derivation∗-commuting derivationinvolutionprime idealprime ring |
| spellingShingle | Shakir Ali Turki M. Alsuraiheed Mohammad Salahuddin Khan Cihat Abdioglu Mohammed Ayedh Naira N. Rafiquee Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications Mathematics derivation ∗-centralizing derivation ∗-commuting derivation involution prime ideal prime ring |
| title | Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications |
| title_full | Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications |
| title_fullStr | Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications |
| title_full_unstemmed | Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications |
| title_short | Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications |
| title_sort | posner s theorem and ∗ centralizing derivations on prime ideals with applications |
| topic | derivation ∗-centralizing derivation ∗-commuting derivation involution prime ideal prime ring |
| url | https://www.mdpi.com/2227-7390/11/14/3117 |
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