preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202306.0163.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 2 June 2023 / Approved: 2 June 2023 / Online: 2 June 2023 (08:59:17 CEST)

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 is central, then the ring is commutative. In the present paper, we extend this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we establish *-version of Vukman’s theorem [48, Theorem 1]. 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-trosion free semiprime ring A with involution admits a nonzero *-centralizing derivation, then A contains a nonzero central ideal. This result is in a spirit of the classical result due to Bell and Martindale [19, Theorem 3]. As the applications, we extends and unify several classical theorems proved in [6],[25,[42], and [48] . Finally, we conclude our paper with a direction for further research.

Derivation; *-centralizing derivation; *-commuting derivation; involution; prime ideal; prime ring; semiprime ring

Computer Science and Mathematics, Algebra and Number Theory

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