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On (reverse) \((\alpha,\beta,\gamma)\)-derivations of associative algebras. (English) Zbl 1369.16023
In this paper, the author describes all \((\alpha,\beta,\gamma)\)-derivations of semiprime associative algebras. He generalizes the notion of reverse derivation and introduces the notion of reverse \((\alpha,\beta,\gamma)\)-derivations. Also, he describes all reverse \((\alpha,1,1)\)-derivations of prime associative algebras. If we go back to the history of the subject reverse derivation, we remark that I. N. Herstein [Proc. Am. Math. Soc. 7, 1021–1022 (1957; Zbl 0077.04201)] introduced the notion of a reverse derivation as an additive map \(d\) on a ring \(R\) satisfying \(d(yx)=d(x)y+ xd(y)\), for all \(x,y\in R\). He showed that if \(R\) is a prime ring, and \(d\) is a nonzero reverse derivation of \(R\), then \(R\) is a commutative integral domain and \(d\) is a derivation. Later, the result of Herstein was generalized in some papers. This article returns to that subject but with new objects which are \((\alpha,\beta,\gamma)\)-derivations. In fact, the concept of reverse derivations has relations with some generalizations of derivations. The author shows some examples as illustration of some concepts. The author presents his results in the first two sections. In Section 3, he deals with \((\alpha,\beta,\gamma)\)-derivations of semiprime associative algebras, while in Section 4 he focuses on reverse \((\alpha,1,1)\)-derivations of prime associative algebras.
16N60 Prime and semiprime associative rings
16W25 Derivations, actions of Lie algebras
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