Two-Local derivations on associative and Jordan matrix rings over commutative rings

In the present paper we prove that every 2-local inner derivation on the matrix ring over a commutative ring is an inner derivation and every derivation on an associative ring has an extension to a derivation on the matrix ring over this associative ring. We also develop a Jordan analog of the above method and prove that every 2-local inner derivation on the Jordan matrix ring over a commutative ring is a derivation.