Commuting Traces of Biadditive Maps on Invertible Elements |
| |
Authors: | Willian Franca |
| |
Institution: | 1. Instituto de Matemática e Estatística, Universidade de S?o Paulo, S?o Paulo, Brazilwfranca39@yahoo.com |
| |
Abstract: | Let R be a simple unital ring. Under a mild technical restriction on R, we will characterize biadditive mappings G: R2 → R satisfying G(u, u)u = uG(u, u), and G(1, r) = G(r, 1) = r for all unit u ∈ R and r ∈ R, respectively. As an application, we describe bijective linear maps θ: R → R satisfying θ(xyx?1y?1) = θ(x)θ(y)θ(x)?1θ(y)?1 for all invertible x, y ∈ R. This solves an open problem of Herstein on multiplicative commutators. More precisely, we will show that θ is an isomorphism. Furthermore, we shall see the existence of a unital simple ring R′ without nontrivial idempotents, that admits a bijective linear map f: R′ → R′, preserving multiplicative commutators, that is not an isomorphism. |
| |
Keywords: | Commuting traces of biadditive maps Functional identities Idempotents Multiplicative commutators Simple rings |
|
|