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

Abstract

Let R be a commutative principal ideal domain (PID) with char ( R) ≠ 2, n ⩾ 2. Denote by S n ( R ) the set of all n × n symmetric matrices over R. If ϕ is a Jordan automorphism on S n ( R ) , then ϕ is an additive rank preserving bijective map. In this paper, every additive rank preserving bijection on S n ( R ) is characterized, thus ϕ is a Jordan automorphism on S n ( R ) if and only if ϕ is of the form ϕ(X) = α t PX σ P where α ∈ R ∗, P ∈ GL n ( R) which satisfies t PP = α −1 I, and σ 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 ϕ is a Jordan automorphism on S n ( R ) if and only if ϕ is an additive rank preserving bijection on S n ( R ) which satisfies ϕ( I) = I.