On additive isomorphisms of prime rings preserving polynomials
A Lie (Jordan, respectively) isomorphism between (associative) rings is an additive isomorphism that preserves a polynomial \(xy-yx\) (\(xy+yx\), respectively). The paper considers additive isomorphisms between prime rings that preserve any multilinear polynomial in noncommuting variables. This of course covers and unifies both the classical cases of Lie and Jordan isomorphisms, as well as some other special cases treated in the literature. A simplified version of the main result is: Let \(B\) and \(A\) be prime rings, \(C\) be the extended centroid of \(A\) and \(F\) be a subring of \(C\) with 1 such that \(B\) is an \(F\)-algebra. Further, let \(m\geq 2\) and \(f(x_1,\ldots,x_m)\in F\langle X\rangle\), the free \(F\)-algebra on the set \(X=\{x_1,x_2,\ldots\}\), be a nonzero multilinear polynomial. Suppose that \(\gamma\colon B\to A\) is an isomorphism of \(F\)-modules such that \(\gamma(f(b_1,\ldots,b_m))=f(\gamma(b_1),\ldots,\gamma(b_m))\) for all \(b_1,\ldots,b_m\in B\). If \(\text{char}(A)\neq 2\) and \(A\) contains elements that are not algebraic of degree \(\leq 2m-1\) over the extended centroid, then \(\gamma\) is of the form \(\gamma(b)=t\beta(b)+\nu(b)\), where \(t\in C\), \(t\neq 0\), \(\beta\) is an (anti-)monomorphism of \(B\) into the central closure of \(A\) and \(\nu\) is an additive map of \(B\) into \(C\). The proof rests heavily on some recent progress made in the theory of functional identities.
scientific article; zbMATH DE number 1340565
scientific article