Jordan isomorphisms and additive rank preserving maps on symmetric matrices over PID

Let R be a commutative principal ideal domain (PID) with char(R) not equal 2, n >= 2. Denote by S-n(R) the set of all n x n symmetric matrices over R. If p is a Jordan automorphism on S-n(R), then phi is an additive rank preserving bijective map. In this paper, every additive rank preserving bijection on S-n(R) is characterized, thus phi is a Jordan automorphism on S-n(R) if and only if phi is of the form phi(X) = alpha(t) PX sigma P where alpha is an element of R*, P is an element of GL(n) (R) which satisfies P-t P = alpha I-1, and a is an automorphism of R. It follows that every Jordan automorphism on S-n(R) may be extended to a ring automorphism on M-n(R), and phi is a Jordan automorphism on S-n(R) if and only if phi is an additive rank preserving bijection on S-n(R) which satisfies phi(I) = I. (c) 2006 Elsevier Inc. All rights reserved.