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 Sn(R) the set of all n × n symmetric matrices over R. If ϕ is a Jordan automorphism on Sn(R), then ϕ is an additive rank preserving bijective map. In this paper, every additive rank preserving bijection on Sn(R) is characterized, thus ϕ is a Jordan automorphism on Sn(R) if and only if ϕ is of the form ϕ(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 Sn(R) may be extended to a ring automorphism on Mn(R), and ϕ is a Jordan automorphism on Sn(R) if and only if ϕ is an additive rank preserving bijection on Sn(R) which satisfies ϕ(I) = I.