Noetherian Ring - Properties

Properties

• If R is a Noetherian ring, then R is Noetherian by the Hilbert basis theorem. Also, R], the power series ring is a Noetherian ring.
• If R is a Noetherian ring and I is a two-sided ideal, then the factor ring R/I is also Noetherian.
• Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian. (This follows from the two previous properties.)
• Every localization of a commutative Noetherian ring is Noetherian.
• A consequence of the Akizuki-Hopkins-Levitzki Theorem is that every left Artinian ring is left Noetherian. Another consequence is that a left Artinian ring is right Noetherian if and only if right Artinian. The analogous statements with "right" and "left" interchanged are also true.
• A ring R is left-Noetherian if and only if every finitely generated left R-module is a Noetherian module.
• A left Noetherian ring is left coherent and a left Noetherian domain is a left Ore domain.
• A ring is (left/right) Noetherian if and only if every direct sum of injective (left/right) modules is injective. Every injective module can be decomposed as direct sum of indecomposable injective modules.