Noetherian and Artinian conditions prevent infinite complexity. They turn infinite algebraic objects into objects controlled by finite generation or finite descent.
Ascending and descending chain conditions
Definition. An \(R\)-module \(M\) is Noetherian if every ascending chain of submodules stabilizes. It is Artinian if every descending chain of submodules stabilizes.
Example. A finite-dimensional vector space is both Noetherian and Artinian, because subspace dimensions cannot increase or decrease indefinitely.
Finite generation criterion
Theorem. An \(R\)-module \(M\) is Noetherian iff every submodule of \(M\) is finitely generated.
Proof. If some submodule is not finitely generated, choose elements successively that are not generated by the previous ones; this gives a strictly increasing chain. Conversely, if every submodule is finitely generated, the union of an ascending chain is generated by finitely many elements, all of which occur in one stage, so the chain stabilizes.
Noetherian rings
Definition. A ring \(R\) is left Noetherian if it is Noetherian as a left module over itself. For commutative rings, this means every ideal is finitely generated.
Example. Every PID is Noetherian because every ideal is generated by one element. Thus \(\mathbb Z\) and \(F[x]\) are Noetherian.
Hilbert basis theorem
Theorem. If \(R\) is a commutative Noetherian ring, then \(R[x]\) is Noetherian. Hence \(R[x_1, \ldots,x_n]\) is Noetherian for every finite \(n\).
Remark. This theorem is the algebraic reason polynomial ideals in finitely many variables have finite generating sets, even when finding such generators is difficult.
Artinian rings and modules
Theorem. An Artinian integral domain is a field.
Proof. For nonzero \(a\in R\), the chain \((a)\supset(a^2)\supset(a^3)\supset\cdots\) stabilizes. Thus \((a^n)=(a^{n+1})\) for some \(n\), so \(a^n=ra^{n+1}\). Since the domain has no zero divisors, \(1=ra\), and \(a\) is invertible.
Examples and non-examples
Example. \(\mathbb Z\) is Noetherian but not Artinian, because \[ (2)\supset(4)\supset(8)\supset\cdots \] is a strictly descending chain of ideals.
Counterexample. The polynomial ring \(F[x_1,x_2,x_3,\ldots]\) in infinitely many variables is not Noetherian, since \[ (x_1)\subset(x_1,x_2)\subset(x_1,x_2,x_3)\subset\cdots \] is a strictly ascending chain of ideals.
Why chain conditions matter
Remark. Noetherian hypotheses let proofs use maximal counterexamples and induction on ideals. Artinian hypotheses force finite decomposition and often reduce rings to field-like pieces.