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?

Good luck explaining $\sqrt{2}\cdot \sqrt{3}$ with addition alone...
So no, you can't just derive the multiplication from the addition. To give you a smaller example, consider the following two rings:


*

*$R = \mathbb{F}_2[x]/\langle x^2\rangle = \{0,1,x,x+1\}$

*$S = \mathbb{F}_2[x]/\langle x^2+x+1\rangle = \{0,1,x,x+1\} \cong \mathbb{F}_4$


In both cases, you have the exact same addition, so you can't tell the difference between these two rings while only looking at addition. But they are of course different, as the second one is a field and the first one is not.
