# nLab rational function

Given a commutative ring $R$, the commutative ring of rational functions with coefficients in $R$ is the field of fractions of the polynomial ring $R\left[z\right]$.

Let $X$ be an affine variety over a field $k$ with the ring of regular function?s $𝒪\left(X\right)$. A rational function is any element of the field of fractions of $𝒪\left(X\right)$, that is the function field of the variety.

In either case, rational functions are equivalence classes of fractions; they need not be functions defined everywhere.

Revised on December 9, 2009 05:17:22 by Toby Bartels (173.60.119.197)