Elliptic Curves over Rings? 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 considered to some extent in this.
http://user.math.uzh.ch/fontein/diplom-fontein.pdf
 A: A commutative ring, yes.  This is treated to some extent in Silverman's second book; for the more general story of "abelian schemes," which is what you're really after, I might look at Milne's articles in the volume Arithmetic Geometry edited by Cornell and Silverman.
As for noncommutative rings, I'm afraid I have no idea -- I'm not even sure what "construction" would be in a position to fail!
A: Elliptic curves can be defined over arbitrary base schemes $S$. In particular, for every (commutative!) ring $R$ one can talk about elliptic curves over (the spectrum of) $R$. Loosely speaking, what one gets is a family $E$ of elliptic curves parametrized by the points of $S$. One then proves the existence of the group law ($E$ can be given the structure of an $S$-group scheme), and goes from there. E.g., locally over $S$, $E$ can be put into Weierstrass form.
In the book Arithmetic Moduli of Elliptic Curves by Katz and Mazur, an elliptic curve over $S$ is defined as a proper smooth morphism $f : E \rightarrow S$ of finite presentation, with a section $0 : S \rightarrow E$, such that all geometric fibers of $f$ are integral (equivalently, connected) curves of genus one.
What can be done for noncommutative $R$ I don't know. It seems to me that you have to say what you mean by an elliptic curve over a noncommutative ring. One can't simply replace 'field' in 'elliptic curve over a field' by the name of some other algebraic structure and expect it to make sense, I guess.
