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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Bibliographic Details
Published in:Mathematics (Basel) 2023-07, Vol.11 (14), p.3117
Main Authors: Ali, Shakir, Alsuraiheed, Turki M., Khan, Mohammad Salahuddin, Abdioglu, Cihat, Ayedh, Mohammed, Rafiquee, Naira N.
Format: Article
Language:English
Subjects:
centralizing derivation
Commutators
Commuting
commuting derivation
Derivation
Food science
involution
prime ideal
prime ring
Theorems
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math11143117