On left derivations and related mappings

Abstract

Let $R$ be a ring and $X$ be a left $R$-module. The purpose of this paper is to investigate additive mappings ${D_1}:R \to X$ and ${D_2}:R \to X$ that satisfy ${D_1}(ab) = a{D_1}(b) + b{D_1}(a),a,b \in R$ (left derivation) and ${D_2}({a^2}) = 2a{D_2}(a),a \in R$ (Jordan left derivation). We show, by the rather weak assumptions, that the existence of a nonzero Jordan left derivation of $R$ into $X$ implies $R$ is commutative. This result is used to prove two noncommutative extensions of the classical Singer-Wermer theorem.