Multiplication is just repeated addition, so we can derive multiplication from addition.

Addition over integers is a monoid. Addition and multiplication over integers form a Ring, so there's one example where this is possible, but addition and multiplication have more properties, for example addition is commutative. I'm not sure if it's possible to derive the identity element of multiplication from the Monoid (in the general ring-sense of multiplication and addition).

Is it possible to derive a Ring from a Monoid in general?