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.