Jordan isomorphisms of upper triangular matrix rings. (Q996226)

A bijective additive map of rings \(g\colon A\to B\) is a Jordan isomorphism if for all \(x,y\in A\), \(g(xy+yx)=g(x)g(y)+g(y)g(x)\). The authors generalize results in the literature that show when such a map is a ring isomorphism or anti-isomorphism. They consider a Jordan isomorphism \(g\colon S\to B\) for \(S\) the ring of upper triangular \(n\times n\) matrices, with \(n\geq 2\), over a 2-torsion free ring \(R\) with identity \(1_R\). The main result of the authors shows that \(g\) must be a sum of an isomorphism and an anti-isomorphism. Specifically, \(R\) contains a central idempotent \(e\) so that \(g\) restricted to \(eS\) is a ring isomorphism, and \(g\) restricted to \((1_R-e)S\) is a ring anti-isomorphism.