Composition and orthogonality of derivations with multilinear polynomials in prime rings
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Composition and orthogonality of derivations with multilinear polynomials in prime rings Balchand Prajapati1
· Charu Gupta1
Received: 11 April 2019 / Accepted: 20 November 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract Let R be a non commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C. Let d and δ be two derivations of R and S be the set of evaluations of a multilinear polynomial f (x1 , . . . , xn ) over C which is not central valued. Let p, q ∈ R. We prove the followings. (1) If pudδ(u) + δd(u)uq = 0 for all u ∈ S and p + q ∈ / C. Then either d = 0 or δ = 0. (2) If pud(u) + d(u)uq = 0 for all u ∈ S. Then either d = 0 or p = q ∈ C, d(x) = [a, x] for some a ∈ U and f (x1 , . . . , xn )2 is central valued. Keywords Prime ring · Derivation · Orthogonal derivations · Extended centroid · Utumi quotient ring Mathematics Subject Classification 16W25 · 16N60
1 Introduction Throughout the article we consider R to be an associative ring. A ring R is said to be prime if for x, y ∈ R and x Ry = 0 implies either x = 0 or y = 0. In this article U denotes the Utumi ring of quotient of a prime ring R and C denotes the center of U . The ring U is prime with unity containing R and C is a field which is known as the extended centroid of R (for more detail see [2]). An additive mapping d : R → R is said to be a derivation if d(x y) = d(x)y + xd(y) for all x, y ∈ R. For a ∈ R, the mapping d(x) = [a, x] for all x ∈ R, is a derivation, which is said to be an inner derivation induced by a. If a derivation d is not an inner derivation it is called an outer derivation. By [24, Theorem 2] we know that every derivation d on a dense right ideal of R can be uniquely extended to U . For b ∈ U if d(x) = [b, x] for all x ∈ R then d is said to be X -inner derivation otherwise X -outer derivation.
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Balchand Prajapati [email protected] Charu Gupta [email protected]
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School of Liberal Studies, Ambedkar University, Delhi 110006, India
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B. Prajapati, C. Gupta
In 1957, Posner [28] proved that if d is a derivation on a prime ring R such that [d(x), x] ∈ Z (R) for all x ∈ R then either d = 0 or R is a commutative ring. Further more generalization of Posner’s result can be found in several ways (see [8,9,27] where further references can be found). In fact, it was the starting point of number of research article concerning the study of various kind of additive mappings satisfying appropriate algebraic conditions on some subset of a prime or semiprime ring R. In [17] Hvala extended the Posner’s result by replacing derivations with generalized derivations. Posner’s theorem also has been studied for composition of derivations (generalized derivations) and derivations (generalized derivations) of order 2 (see [9,16,29,30] where more references can be found). Recently more results on composition of generalized derivations on prime ring have been found in [11–13]. In this paper we prove the following result which describes the identity related to annihila
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