noether's work "abstrakter aufbau der idealtheorie in algebraischen zahl- und funktionenkörpern" ("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-dimensional, and integrally closed in their quotient fields. this paper also contains what now are called the isomorphism theorems, which describe some fundamental natural isomorphisms, and some other basic results on noetherian and artinian modules.