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An Artinian ring satisfies the descending chain condition on ideals. A commutative ring is Artinian exactly when it is Noetherian and has <Krull dimension> zero.

Artinian ring

For a prime ideal $\mathfrak p$, its height is the supremum of lengths $r$ of strict chains
$$
\mathfrak p_0\subsetneq\mathfrak p_1\subsetneq\cdots\subsetneq\mathfrak p_r=\mathfrak p.
$$
The <height of an ideal> $I$, without assuming $I$ prime, is
$$
\boxed{\operatorname{ht}(I)=
\inf_{\mathfrak p\supseteq I}\operatorname{ht}(\mathfrak p)}.
$$
The <Krull dimension> of a ring $R$ is
$$
\boxed{\dim R=\sup\{r:\mathfrak p_0\subsetneq\cdots\subsetneq\mathfrak p_r
\text{ are prime ideals of }R\}}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 4 / i / Solution

The denominator has degree one, independently of the number $n$ of variables. This reflects the fact that all mixed monomials vanish and each component of the ring supports only one polynomial direction; equivalently, the <Krull dimension> of this graded ring is one.

Solved by gpt-5.6-sol high.

Paper 101 / 5 / c / ii / Solution

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The Krull dimension of a commutative ring is the supremum of the lengths of strict chains of prime ideals.


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 Incoming links: Krull dimension