= Ring of rational functions on a reduced variety
{title2=$\operatorname{Rat}(X)$}
For a possibly reducible <algebraic variety> $X$, take pairs $(U,f)$ with $U$ a dense open subset and $f$ a <regular function> on $U$. Identify pairs agreeing on a common dense open subset. Addition and multiplication on intersections make these classes a <ring>. For an <affine variety> with reduced <coordinate ring> $A$, this ring is its <total quotient ring>. For an <irreducible variety>, it is the <function field>. An <isomorphism> of dense open subsets induces an <isomorphism> of these rings.
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