Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the definition of excellent ring.
42.
It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules.
43.
The integral closure of a Dedekind domain in a finite extension of the field of fractions is a Dedekind domain; in particular, a noetherian ring.
44.
The Jaffard property is satisfied by any Noetherian ring " R ", so examples of non-Jaffardian rings are quite difficult to find.
45.
A nonzero finitely generated module " M " over a commutative Noetherian ring is coprimary if and only if it has exactly one associated prime.
46.
Jean-Pierre Serre defined a regular ring as a commutative noetherian ring of finite global homological dimension and proved that this is equivalent to the definition above.
47.
The Lasker Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally the fundamental theorem of finitely generated abelian groups to all Noetherian rings.
48.
He established the concept of the Krull dimension of a ring, first for Noetherian rings before moving on to expand his theory to cover general valuation rings and Krull rings.
49.
The converse is true for finitely generated modules over Noetherian rings : a finitely generated module over a commutative noetherian ring is locally free if and only if it is projective.
50.
The converse is true for finitely generated modules over Noetherian rings : a finitely generated module over a commutative noetherian ring is locally free if and only if it is projective.
