 Ring of rational functions on a reduced variety

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.