Local Mappings Generated by Multiplication on Rings of Matrices

In the present paper, the notion of (additive) local multiplier on the ring of matrices over an arbitrary unital associative ring is introduced and investigated. It is proved that every local left multiplier on the matrix ring over a division ring is a left multiplier. Also, in the present paper, the notion of (additive) local Jordan multiplier on a Jordan ring of symmetric matrices is introduced and investigated. It is proved that every local Jordan multiplier on the Jordan ring of symmetric matrices over the field of rational numbers is a Jordan multiplier. Also corollaries of these results are proved.