On a classical theorem of Noether in ideal theory

Abstract

A classical result in the ideal theory of commutative rings is that an integral domain D with unit is a Dedekind domain if and only if D is noetherian, of dimension less than two, and integrally closed. [8; 275]. The statement of this theorem is due essentially to Noether [6; 53], though the present statement is a refined version of Noether's theorem. (See Cohen [1; 32] for the historical development of the theorem above.) Noether did not, in fact, require that the domain D contain a unit element. By imposing greater restrictions on the prime ideal factorization of each ideal, she showed that D must contain a unit element.