Jordan homomorphisms, chain geometries and the fundamental theorem (Q921581)

The author asks what conditions force a Jordan homomorphism of K-algebras to define a homomorphism of chain geometries. He answers in the case of the stable rank 2. He gives a condition, equivalent to the K- semilinearity when K is central, so that a Jordan homomorphism of K- algebra induces a homomorphism of the chain geometries by its direct action on the coordinates of a point. The hypothesis of stable rank 2 for a ring comes from K-theory and is satisfied by a reasonably large class of rings including every algebra of finite dimension. In the first part of this article a few properties of Jordan homomorphisms of arbitrary rings are proved. In the second section the above mentioned result is established.