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principal divisor
Let \(X\) be a noetherian integral separated scheme which is regular in codimension one and let \(Y\) be a prime divisor with generic point \(\eta\). Then \({\mathcal{O}}_{X, \eta}\) is a DVR with residue field \(K\). There is a map
\begin{align*}
K^{\times}\to \operatorname{Div}(X) \\
f \mapsto (f) \coloneqq\sum_{y\in \operatorname{Div}(X)} v_y(f)\, y
,\end{align*}
which is well-defined since infinitely many \(v_y(f) = 0\) since the non-regular locus of \(f\) is a proper closed subset of a Noetherian scheme, and thus contains only finitely many prime divisors. Any divisor in the image is called prinicipal. 