Jordan maps on triangular algebras. (Q996229)

For \(x,y\in R\), an associative ring, let \(x\circ y=xy+yx\). Given rings \(R\) and \(S\), maps \(f\colon R\to S\) and \(g\colon S\to R\) form a Jordan pair if for all \(x\in R\) and \(y\in S\), \(f(x\circ g(y))=f(x)\circ y\) and \(g(y\circ f(x))=g(y)\circ x\). The author shows that in certain cases each of \(f\) and \(g\) forming a Jordan pair must be additive maps. Necessary conditions for this conclusion are that both \(f\) and \(g\) are surjective, and that one of \(R\) or \(S\) is a faithful triangular algebra. Such an algebra is a \(2\times 2\) triangular matrix ring with upper and lower diagonal entries from algebras \(A\) and \(B\), respectively, over a commutative ring \(R\), so that \(a\circ A=0\) for \(a\in A\) forces \(a=0\) and \(b\circ B=0\) for \(b\in B\) forces \(b=0\), and with the upper right entries coming from a faithful \((A,B)\) module \(X\) so that neither \(A\) nor \(B\) annihilates any nonzero \(x\in X\).