Noetherian RingEdit

Noetherian rings are a cornerstone of modern algebra, providing a robust finiteness framework that keeps many problems from becoming unwieldy. The concept, named after Emmy Noether, sits at the heart of commutative algebra and algebraic geometry, and it has natural extensions to modules and various noncommutative settings. In a Noetherian ring, the structure of ideals is tightly controlled: every ideal is generated by finitely many elements, and ascending chains of ideals must eventually stabilize. This finiteness leads to powerful theorems and a cleaner geometric picture of the algebra in play.

Definition and basic properties

A ring R is called Noetherian if it satisfies the ascending chain condition (ACC) on ideals: every increasing sequence of ideals I1 ⊆ I2 ⊆ I3 ⊆ … eventually stabilizes. In the noncommutative case, one can speak of left-Noetherian (ACC on left ideals) or right-Noetherian (ACC on right ideals) rings; for commutative rings, these notions coincide.
Equivalently, R is Noetherian if every ideal of R is finitely generated. This is the most commonly used characterization in commutative algebra.
If R is Noetherian and S is a quotient ring R/I, then S is Noetherian. Similarly, if R is Noetherian and A is a finitely generated R-algebra, then A is Noetherian. These consequences often follow from Hilbert’s Basis Theorem (see below).
The property is stable under several standard constructions: finite products of Noetherian rings are Noetherian, localizations at multiplicative sets preserve Noetherianness, and polynomial rings in finitely many variables over a Noetherian ring are Noetherian.

Among the most important equivalent characterizations is that every ascending chain of ideals stabilizes, which provides a practical criterion for proving that a given ring is Noetherian.

Noetherian rings in practice: key results

Hilbert Basis Theorem: If R is Noetherian, then the polynomial ring R[x1, x2, ..., xn] is Noetherian. Consequently, any finitely generated algebra over a Noetherian ring is Noetherian.
Quotients: If R is Noetherian and I ⊆ R is an ideal, then R/I is Noetherian.
Finite generation of ideals: In a Noetherian ring, every ideal can be described by finitely many generators.
Local properties: If R is Noetherian, then so is its localizations R_S at any multiplicative subset S.
Geometry meets algebra: The spectrum of a Noetherian ring, denoted Spec(R), is a Noetherian topological space. This aligns algebraic finiteness with geometric finiteness in algebraic geometry.
Primary decomposition: In a Noetherian ring, every ideal has a primary decomposition. This provides a structured way to analyze the radical and associated primes of ideals.
Noether normalization: For a finitely generated algebra A over a field k, there exists a polynomial subring over which A is integral. This result is a bridge between finiteness properties and geometric dimension in algebraic geometry.

Examples and non-examples

The ring of integers, Z, is Noetherian. Every ideal (n) is finitely generated, and ascending chains stabilize.
If k is a field, the polynomial ring k[x1, ..., xn] is Noetherian for any finite n, by Hilbert’s Basis Theorem.
Quotients of Noetherian rings are Noetherian. For example, k[x]/(f) is Noetherian for any nonconstant polynomial f ∈ k[x].
Finite products of Noetherian rings are Noetherian.
Localization preserves Noetherianness: if R is Noetherian and S is a multiplicative set, then R_S is Noetherian.
Rings that fail the ACC on ideals provide classic non-examples. For instance, the polynomial ring in infinitely many variables over a field, k[x1, x2, x3, ...], is not Noetherian. This contrasts with the finite-variable case and highlights the role of finiteness in the theory.

Noetherian modules and related structure

A module M over a ring R is called Noetherian if it satisfies the ascending chain condition on submodules: every increasing sequence of submodules N1 ⊆ N2 ⊆ N3 ⊆ … stabilizes.
Equivalent characterizations exist: every submodule of M is finitely generated, or every submodule is a finite union of certain manageable pieces in specific contexts.
If R is Noetherian and M is a finitely generated R-module, then M is a Noetherian R-module. The converse is not automatic, but finitely generated modules over Noetherian rings retain a great deal of finiteness.
Noetherian modules underpin many constructions in algebraic geometry and representation theory, providing a controlled setting for decompositions and support considerations.

Connections to algebraic geometry and topology

The spectrum of a Noetherian ring, Spec(R), is a Noetherian topological space. This makes the geometry of algebraic varieties over Noetherian rings tractable and well-behaved.
Noetherian hypotheses underpin many fundamental theorems about dimensions, fibers, and morphisms in algebraic geometry. They also facilitate the study of coherent sheaves and the fibration properties of schemes.
In broad terms, Noetherianity ensures that many geometric and algebraic processes terminate after finitely many steps, which is crucial for effective computation and structural theorems.