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It is known that we can define elliptic curves over commutative rings. However can we define an elliptic curve over a noncommutative ring?

This question is considered to some extent in this thesis (Section 4.4) but reaches no conclusion on the matter.

NOTE: By an elliptic curve over a noncommutative ring I do not mean a noncommutative torus.

I asked a similar question on mathoverflow before but got no replies in the affirmative or the negative.

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up vote 6 down vote accepted

There is no generally accepted or straightforward definition of a curve, or more generally a variety, over non-commutative rings. Noncommutative geometry is an ongoing subject of research, but the basic definitions and constructions are not settled, unlike in the situation of algebraic geometry (which takes place over commutative rings). If you want to know why, you could look at this answer, and related material on non-commutative geometry posted on MO.

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Thank you for the very helpful answer! –  Eugene Jun 7 '12 at 13:05

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