Identities with inverses on matrix rings
Journal
Linear and Multilinear Algebra
Journal Volume
68
Journal Issue
3
Pages
635-651
Date Issued
2020
Author(s)
Abstract
Motivated by [1, 2], the goal of the paper is to study certain identities with inverses on matrix rings. Given D a division ring, we characterize additive maps f, g: D → D satisfying the identity f(x)x−1 + xg(x−1) = 0 for all invertible x ∈ D. Let R be a matrix ring over a division ring of characteristic not 2. We also characterize additive maps f, g: R → R satisfying the identity f(x)x−1 + xg(x−1) = 0 for all invertible x ∈ R. Precisely, there exist an element q ∈ R and a derivation d of R such that f(x) = xq + d(x) and g(x) = −qx + d(x) for all x ∈ R. This affirmatively answers the question below Theorem 4 in [1] due to L. Catalano.
Type
journal article