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

It depends on what you want to do.
For any monoid $M$ there is alway the monoid ring $\Bbb Z[M]$. This is the left adjoint to the forgetful functor from rings to monoids.
If you want to take a commutative monoid as ground for the underlying group of the ring you probably want to take the of its Grothendieck group before. But this could be any abelian group. And if the ring is required to have a multiplicative identity element this can not always be achieved
