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.

中文翻译：

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矩阵环乘法生成的局部映射

在本文中，引入并研究了任意单位关联环上矩阵环上的（加性）局部乘子的概念。证明了除法环上矩阵环上的每个局部左乘子都是左乘子。此外，在本论文中，介绍并研究了对称矩阵 Jordan 环上的（加性）局部 Jordan 乘子的概念。证明了有理数域上对称矩阵的乔丹环上的每个局部乔丹乘子都是乔丹乘子。这些结果的推论也得到了证明。