zbMATH — the first resource for mathematics

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.
MSC:
 16N60 Prime and semiprime associative rings 16W25 Derivations, actions of Lie algebras
Full Text:
References:
 [1] Aboubakr, A; González, S, Reverse generalized derivations on semiprime rings, Sib. Math. J., 56, 199-205, (2015) · Zbl 1335.16031 [2] Benkovic, D, Jordan derivations and antiderivations on triangular matrices, Linear Algebra Appl., 397, 235-244, (2005) · Zbl 1072.15021 [3] Bresar, M; Vukman, J, Jordan derivations on prime rings, Bull. Aust. Math. Soc., 37, 321-322, (1988) · Zbl 0634.16021 [4] Bresar, M., Vukman, J.: Jordan $$(θ ,ϕ )$$-derivations. Glas. Mat. Ser. III 26(46), 1-2, 13-17 (1991) · Zbl 0798.16023 [5] Burde, D; Dekimpe, K, Post-Lie algebra structures and generalized derivations of semisimple Lie algebras, Moscow Math. J., 13, 1-18, (2013) · Zbl 1345.17011 [6] Filippov, VT, On $$δ$$-derivations of Lie algebras, Sib. Math. J., 39, 1218-1230, (1998) · Zbl 0936.17020 [7] Filippov, VT, $$δ$$-derivations of prime Lie algebras, Sib. Math. J., 40, 174-184, (1999) · Zbl 0936.17021 [8] Filippov, VT, On $$δ$$-derivations of prime alternative and malcev algebras, Algebra Log., 39, 354-358, (2000) [9] Herstein, I, Jordan derivations of prime rings, Proc. Am. Math. Soc., 8, 1104-1110, (1957) · Zbl 0216.07202 [10] Hopkins, NC, Generalized derivations of nonassociative algebras, Nova J. Math. Game Theory Algebra, 5, 215-224, (1996) · Zbl 0883.17005 [11] Kaygorodov, I, On $$δ$$-derivations of simple finite-dimensional Jordan superalgebras, Algebra Log., 46, 318-329, (2007) · Zbl 1164.17020 [12] Kaygorodov, I, On $$δ$$-derivations of classical Lie superalgebras, Sib. Math. J., 50, 434-449, (2009) · Zbl 1221.17019 [13] Kaygorodov, I, $$δ$$-superderivations of simple finite-dimensional Jordan and Lie superalgebras, Algebra Log., 49, 130-144, (2010) · Zbl 1232.17028 [14] Kaygorodov, I, On $$δ$$-superderivations of semisimple finite-dimensional Jordan superalgebras, Math. Notes, 91, 187-197, (2012) · Zbl 1376.17043 [15] Kaygorodov, I, On $$(n+1)$$-ary derivations of simple $$n$$-ary malcev algebras, St. Petersb. Math. J., 25, 575-585, (2014) · Zbl 1380.17025 [16] Kaygorodov, I, Jordan $$δ$$-derivations of associative algebras, J. Gen. Lie Theory Appl., S1, 003, (2015) · Zbl 1371.16026 [17] Kharchenko, V.: Noncommutative Galois Theory [in Russian]. Nauch. Kniga, Novosibirsk (1996) · Zbl 0974.12500 [18] Novotny, P; Hrivnak, J, On $$(α, β, γ )$$-derivation of Lie algebras and corresponding invariant functions, J. Geom. Phys., 58, 208-217, (2008) · Zbl 1162.17014 [19] Samman, M; Alyamani, N, Derivations and reverse derivations in semiprime rings, Int. Math. Forum, 2, 1895-1902, (2007) · Zbl 1146.16016 [20] Vukman, J, An identity related to centralizers in semiprime rings, Comment. Math. Univ. Carol., 40, 447-456, (1999) · Zbl 1014.16021 [21] Zheng, K., Zhang, Y.; On $$(α ,β ,γ )$$-derivations of Lie superalgebras. Int. J. Geom. Methods Mod. Phys. 10, 10, 1350050 (2013) (18 pages) · Zbl 1370.17023 [22] Zheng, K; Zhang, Y, A note on $$(α ,β ,γ )$$-superderivations of superalgebras, J. Math. Res. Appl., 34, 627-630, (2014) · Zbl 1324.17002 [23] Zusmanovich, P, On $$δ$$-derivations of Lie algebras and superalgebras, J. Algebra, 324, 3470-3486, (2010) · Zbl 1218.17011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.