Affine algebraic curve
= Affine algebraic curve
= Affine curve
{synonym}
An affine algebraic curve is an <affine variety> of <Krull dimension> one. An irreducible affine curve has a finitely generated one-dimensional domain as its <coordinate ring>. Quotienting this ring by a nonzero element gives a zero-dimensional <Noetherian ring>, hence an <Artinian ring>. Consequently a nonzero regular function has finitely many zeros. No nonsingularity assumption is part of the definition.