Adjective
(algebra) Of a ring in which any ascending chain of ideals eventually starts repeating.
(algebra) Of a module in which any ascending chain of submodules eventually starts repeating.
Noetherian (not comparable)
Alternative letter-case form of noetherian
Applications Let be a Noetherian commutative ring. Source: Internet
Bourbaki, 7.3, no 6, Proposition 4. The converse of this is not true: there are Noetherian local rings that are UFDs but whose completions are not. Source: Internet
A valuation ring is not Noetherian unless it is a principal ideal domain. Source: Internet
Another consequence is that a left Artinian ring is right Noetherian if and only if right Artinian. Source: Internet
Equivalent conditions for a ring to be a UFD A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given below). Source: Internet
For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. Source: Internet