Conventions for this development and where dependent choice and Zorn's lemma are used

Rings. Every ring on this page is commutative (Commutative ring) and carries an identity, since Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides builds the identity into the definition of a ring; the word "ring" below means that wherever a statement does not say otherwise. A statement that reads "let $R$ be a ring" with no commutativity hypothesis is proved without using commutativity and applies in particular to the commutative rings this page is about. Ring homomorphisms preserve the identity (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$) and subrings contain the ambient identity (Subring: a subset containing $1_R$ and closed under addition, additive inverses and multiplication). Nothing here requires $1\mathrm{\ne }0$: the zero ring is a ring, it is admitted by every statement that does not exclude it in as many words, and it is Noetherian, its one ideal being $0=R$ and generated by the empty set.

Noetherian and Artinian are cited, not redefined. Left and right Noetherian rings calls $R$ left Noetherian when the left regular module ${}_{R}R$ is Noetherian, and Noetherian modules: every submodule is finitely generated calls a module Noetherian when every submodule of it is finitely generated. For a commutative ring the left and right conditions coincide, so the side is not written below. The descending-chain dual is Left and right Artinian rings, which calls $R$ left Artinian when ${}_{R}R$ is Artinian; no statement on this page assumes or concludes that condition, and no argument below uses it. The equivalence of finite generation with the ascending chain condition and with the maximal condition is Finite generation, ACC, and maximal-condition characterizations of Noetherian modules, proved for modules; the ideal-level form is obtained on this page by transporting that theorem across the identification of the ideals of $R$ with the submodules of ${}_{R}R$, rather than by proving the cycle a second time.

Where a choice principle is used, and where none is. The definition of finite generation, and every argument below that produces a finite generating set from data already in hand, are theorems of ZF. Two places are different, and both are marked where they occur.

• Dependent choice. The implication from the ascending chain condition to the maximal condition is not choice-free: from a nonempty family of ideals with no maximal member one builds a strictly ascending chain by choosing, at each stage, an ideal strictly containing the one already chosen, and the sequence of choices depends on the choices already made. Finite generation, ACC, and maximal-condition characterizations of Noetherian modules attributes exactly that implication to dependent choice, and the ideal-level statement on this page carries the attribution unchanged. The reverse implication, and the two implications relating finite generation to the other conditions, need no choice principle. Sources differ here: Milne §3 attributes the implication to dependent choice and Altman–Kleiman (16.4) to countable choice. The two principles are not the same, and the published module theorem's attribution is the one in force inside this library.
• Zorn's lemma, hence the axiom of choice. Cohen's criterion below is proved by producing an ideal maximal among those that are not finitely generated, in a ring not yet known to be Noetherian. No chain condition is available to supply that maximal element, and it is obtained from Zorn's lemma applied to the set of non-finitely-generated ideals ordered by inclusion. This is a strictly stronger commitment than the dependent choice above, and the lemmas leading to Cohen's criterion say so in their own statements.

What "finitely generated" is measured against. For an ideal it means finite generation as an ideal of the ring, for a module finite generation as a module over the ring named, and for an algebra it means the algebra generated by finitely many elements. These are three different conditions and the ring or module against which each is taken is written out at every occurrence, because a subring of a ring $B$ can be finitely generated as an algebra over $A$ and not as an $A$-module.