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.
