Let \(A\) be an associative ring and \(\nabla :A\to A\) be an additive map. If for each \(x\in A\) there exists an element \(a_x\) (depending on \(x\)) in \(A\) such that \(\nabla (x)=a_xx\) (resp., \(\nabla (x)=xa_x\)), then \(\nabla \) is called a local left (resp., right) multiplier. If there exists an \(a\in A\) such that \(\nabla (x)=ax\) (resp., \(\nabla (x)=xa\)) for all \(x\in A\), then the map \(\nabla \) is called a left (resp., right) multiplier.
Let \(R\) be a unital ring with invertible element 2. An additive map \(\psi :R\to R\) is called a local Jordan multiplier if for each \(x\in R\) there exists an element \(a_x\) (depending on \(x\)) in \(R\) such that \(\psi (x)=\frac{1}{2}(a_xx+xa_x)\). If there exists an element \(a\in R\) such that \(\psi (x)=\frac{1}{2}(ax+xa)\) for all \(x\in R\), then the additive mapping \(\psi \) is called a Jordan multiplier.,
In the paper under review, it is proved that every local left (resp., right) multiplier on the matrix ring over a division ring is a left (resp., right) multiplier. It is also shown that every local Jordan multiplier on the Jordan ring of symmetric matrices over the field of rational numbers is a Jordan multiplier.