A commutative ring is called right-noetherian iff the set of all right ideals of , ordered by inclusion, satisfies the ascending chain condition.
Left-noetherian rings are similarly defined.
Definition (noetherian ring):
A commutative ring is called noetherian iff the set of all ideals of , ordered by inclusion, satisfies the ascending chain condition.
We will state and prove results only for right-noetherian rings, even though they are valid mutatis mutandis for left-noetherian and noetherian rings just as well.
Proposition (ideals of noetherian rings contain powers of their radicals):
Let be a noetherian ring, and let be an ideal. Then there exists such that
Proof: Let be a basis of considered as an -module. Then choose sufficiently large so that
Then define and observe that whenever , then by the pigeonhole principle, upon expanding the expression and considering each summand, we find that for each summand there is at least one so that the corresponding power of is bigger than or equal to .
Proposition (elements of noetherian rings are products of irreducible elements):
Let be a noetherian ring, and let be a non-unit. Then there exist irreducible elements so that
(On the condition of the dependent choice.)
Proof: Indeed, a non-unit factors as , where is a non-unit and either is irreducible and is a unit, or are both irreducible, or is not irreducible and a proper divisor of . The same is true for , and proceeding inductively we gain an ascending chain
which stabilizes by the noetherian assumption. But if is chosen large enough so that the sequence stabilizes after , is irreducible. Hence, we may factor , where is irreducible, and continuing in the same fashion we obtain again an ascending sequence, whose stabilization implies the desired factorisation.