Local mappings generated by a multiplication on rings of matrices

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