Pair of (Generalized-)Derivations on Rings and Banach Algebras

Abstract

Abstract. Let n be a fixed positive integer, R be a 2n!-torsion free prime ring and µ, ν be a pair of generalized derivations on R. If µ 2 (x) + ν(x), x n = 0 for all x ∈ R, then µ and ν are either left multipliers or right multipliers. Let n be a fixed positive integer, R be a noncommutative 2n!-torsion free prime ring with the center C R and d, g be a pair of derivations on R.Then we apply these purely algebraic techniques to obtain several range inclusion results of pair of (generalized-)derivations on a Banach algebra.

Generalized Derivations and Derivations of Rings and Banach Algebras

“…If there exist derivations d, g : R → R such that the mappingx → d 2 (x) + g(x), x m is n-anti-centralizing on R, then we have d = g = 0 on R.Proof. In view of the same process as in the proof of Theorem 2.3, the relation (2.6) yields that d 2 is a derivation on R. From[13], it follows that d = 0 on R and so the relation(2.6) gives g = 0 on R, i.e., d = g = 0 on R. Theorems 2.4 and 2.5 are to improve Corollaries 3.2 and 3.3 of[22], respectively. We continue the next result.Theorem 2.6.…”

“…We also obtain some range inclusion results for generalized linear derivations and linear derivations on Banach algebras by applying the algebraic techniques. Some results in this note are to improve the ones in [22].…”

“…The study of (skew-)centralizing and (skew-)commuting mappings was initiated by a well-known theorem of Posner [13] which states that the existence of a nonzero centralizing derivation on a prime ring implies that the ring is commutative. This theorem has been extended by several authors in different ways ( [21], [22], etc).…”

“…In this note, we investigate anti-centralizing and skew-centralizing mappings involving generalized derivations and derivations on prime rings, semiprime rings and Banach algebras, and some results are to improve the ones in [22].…”