Noether's work " Abstrakter Aufbau der Idealtheorie in algebraischen Zahl-und Funktionenk�rpern " ( " Abstract Structure of the Theory of Ideals in Algebraic Number and Function Fields ", 1927 ) characterized the rings in which the ideals have unique factorization into prime ideals as the Dedekind domains : integral domains that are Noetherian, 0 or 1-integrally closed in their quotient fields.
22.
Then " A " is integrally closed if and only if ( i ) " A " is the intersection of all localizations A _ \ mathfrak { p } over prime ideals \ mathfrak { p } of height 1 and ( ii ) the localization A _ \ mathfrak { p } at a prime ideal \ mathfrak { p } of height 1 is a discrete valuation ring.
23.
Indeed, this is essentially an algebraic translation of these geometric terms : the coordinate ring of any affine variety is, by definition, a finitely generated " k "-algebra, so Noetherian; moreover " curve " means " dimension one " and " nonsingular " implies ( and, in dimension one, is equivalent to ) " normal ", which by definition means " integrally closed ".
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