Noetherian Rings and the ACC

Noetherian rings satisfy a finiteness condition on ideals that ensures all ideal-theoretic constructions terminate, making them the natural setting for commutative algebra and algebraic geometry.

Definition and Equivalences

Definition6.4Noetherian ring

A commutative ring $R$ is Noetherian if it satisfies the ascending chain condition (ACC) on ideals: every ascending chain $I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots$ of ideals eventually stabilizes ( $I_n = I_{n+1} = \cdots$ for some $n$ ).

Theorem6.5Equivalent conditions for Noetherian

The following are equivalent for a commutative ring $R$ :

$R$ is Noetherian (ACC on ideals).
Every ideal of $R$ is finitely generated.
Every nonempty set of ideals has a maximal element (with respect to inclusion).

Key Examples

ExampleNoetherian and non-Noetherian rings

Noetherian:

Any field (ideals: $\{0\}$ and the whole field).
$\mathbb{Z}$ (PID, hence Noetherian).
$k[x_1, \ldots, x_n]$ (by Hilbert's basis theorem).
Any quotient of a Noetherian ring.
$\mathbb{Z}[x]/(x^2 + 1) \cong \mathbb{Z}[i]$ .

Non-Noetherian:

$k[x_1, x_2, x_3, \ldots]$ (polynomial ring in infinitely many variables): the chain $(x_1) \subset (x_1, x_2) \subset \cdots$ never stabilizes.
The ring of all algebraic integers $\overline{\mathbb{Z}}$ .

Primary Decomposition

Theorem6.6Lasker-Noether theorem

In a Noetherian ring, every ideal has a primary decomposition: $I = Q_1 \cap \cdots \cap Q_n$ where each $Q_i$ is primary ( $ab \in Q_i$ and $a \notin Q_i$ implies $b^m \in Q_i$ for some $m$ ). The radicals $\mathfrak{p}_i = \sqrt{Q_i}$ are the associated primes of $I$ .

If the decomposition is irredundant (no $Q_i$ can be removed) and minimal (the $\mathfrak{p}_i$ are distinct), then the set $\{\mathfrak{p}_1, \ldots, \mathfrak{p}_n\}$ is uniquely determined by $I$ .

ExamplePrimary decomposition in $\\mathbb{Z}$

$(12) = (4) \cap (3)$ in $\mathbb{Z}$ : $(4)$ is $(2)$ -primary and $(3)$ is $(3)$ -primary. Associated primes: $\{(2), (3)\}$ .

In $k[x,y]$ : $(x^2, xy) = (x) \cap (x^2, y)$ . Here $(x)$ is prime and $(x^2, y)$ is $(x,y)$ -primary.

RemarkGeometric interpretation

Primary decomposition is the algebraic analog of decomposing a variety into irreducible components (plus embedded components). The associated primes correspond to the irreducible components of $V(I)$ (the minimal primes) and the embedded points (the embedded primes).