For some integral domain $R$, one forms the field of fractions $R^*$ by considering (equivalence classes of) formal pairs {$r/s : r \in R, s\in R\backslash 0$} and defining $+$ and $*$ as you'd expect for fractions.

What happens when you replace $R\backslash 0$ with $R$? Clearly you don't get a field, or even a ring. The result would be a commutative rig that looks a lot like $R^*$, except you also have a sort of "infinity" element $1/0$ and a sort of "undefined" element $0/0$, neither of which have multiplicative *or additive* inverses.

This type of structure seems to arise quite naturally when considering algebras over 1D projective space. So, my questions are

Is this construction well studied, or at least have an accepted name? If so, where is a good starting place w.r.t. relevant literature or results?

`$1/0$`

and`$0/0$`

are the same equivalence class. – Mark Grant Jan 13 '11 at 15:48