So an elliptic curve $E$ over a field $K$ is a smooth projective nonsingular curve of genus $1$ together with a point $O \in E$.

I was reading Silverman's "Arithmetic of Elliptic Curves" and it seems that most of its treatment is over fields.

My question is, does it make sense to define an elliptic curve over a ring (eg: a noncommutative ring)? If not, why not (where would the "construction" fail)? Is it simply not an object of much interest?

Edit: Apparently the question of elliptic curves over noncommutative rings is explored to some extent in this guy's thesis.

Apparently this question is explored to some extent in this guy's thesis.

user.math.uzh.ch/fontein/diplom-fontein.pdf