If is a left module, then an additive map is a left derivation if for all , and is a Jordan left derivation if for all . The main theorem of the paper shows that if is 6-torsion free and no nonzero submodule has an annihilator in , then the existence of a nonzero Jordan left derivation forces to be commutative. One corollary of this result shows that there are no nonzero Jordan left derivations of the algebra of continuous operators on , a Hausdorff locally convex vector space, into either or . When is a left derivation, the torsion assumption in the main theorem can be removed, and also, if , then is central when is a semi-prime ring.
The authors apply their results to a Banach algebra by showing that if is a continuous linear left derivation, then , and if is a continuous linear Jordan derivation with for all , then again . A final application is to functional equations. Let be a Banach space, the algebra of bounded linear operators on , and and additive maps of into either or . If for all invertible , then and for all .