Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications

Shakir Ali (), Turki M. Alsuraiheed, Mohammad Salahuddin Khan, Cihat Abdioglu, Mohammed Ayedh and Naira N. Rafiquee
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Shakir Ali: Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India
Turki M. Alsuraiheed: Department of Mathematics, King Saud University, Riyadh 11495, Saudi Arabia
Mohammad Salahuddin Khan: Department of Applied Mathematics, Z. H. College of Engineering & Technology, Aligarh Muslim University, Aligarh 202002, India
Cihat Abdioglu: Department of Mathematics & Science Education, Karamanoglu Mehmetbey University, Karaman 70100, Turkey
Mohammed Ayedh: Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India
Naira N. Rafiquee: Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India

Mathematics, 2023, vol. 11, issue 14, 1-20

Abstract: 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 A / L , where A is an arbitrary ring and L is a prime ideal of A . Further, by taking advantage of the ∗-version of Vukman’s theorem, we show that if a 2-torsion free semiprime A with involution admits a nonzero ∗-centralizing derivation, then A 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.

Keywords: derivation; ?-centralizing derivation; ?-commuting derivation; involution; prime ideal; prime ring; semiprime ring (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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