Derivations with Invertible Values on a Lie Ideal

Abstract

Let R be a ring which possesses a unit element, a Lie ideal U ⊄ Z, and a derivation d such that d(U) ≠ 0 and d(u) is 0 or invertible, for all u ∈ U. We prove that R must be either a division ring D or D2, the 2 X 2 matrices over a division ring unless d is not inner, R is not semiprime, and either 2R or 3R is 0. We also examine for which division rings D, D2 can possess such a derivation and study when this derivation must be inner.

Derivations on a Lie Ideal

“…We shall make use of the results in [4] and [5] where the authors study derivations with invertible and nilpotent values respectively on a Lie ideal.…”

Derivations with invertible values in flexible algebras

“…In [8] semiprime associative rings with involution, allowing a derivation with invertible values on the set of symmetric elements, were given an examination. In [3] Bergen and Carini studied associative rings admitting a derivation with invertible values on some non -central Lie ideal. Also in the papers [4] and [9] the structure of associative rings that admit α -derivations with invertible values and their natural generalizations -(σ, ?τ ) -derivations with invertible values was described.…”

Superderivations with invertible values

“…Let R be an associative ring. An additive mapping d : R → R satisfying the condition d(xy) = d(x)y + xd(y), for all x, y ∈ R, is called a derivation of R. We say that R has a derivation d with zero or invertible values if d(x) is either zero or invertible for all x ∈ R. In [3], it is proved that a unital ring R possessing a nonzero derivation with zero or invertible values must be isomorphic to either (1) Following this result, many generalizations are obtained in the literature (see, e.g. [1, 2, 4-11, 13, 14]).…”