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

Let R be a commutative principal ideal domain (PID) with char (R) ≠ 2, n 2. Denote by the set of all n × n symmetric matrices over R. If is a Jordan automorphism on , then is an additive rank preserving bijective map. In this paper, every additive rank preserving bijection on is characterized, thus is a Jordan automorphism on if and only if is of the form (X) = αtPXσP where α R*, P GLn(R) which satisfies tPP = α−1I, and σ is an automorphism of R. It follows that every Jordan automorphism on may be extended to a ring automorphism on Mn(R), and is a Jordan automorphism on if and only if is an additive rank preserving bijection on which satisfies (I) = I.