Order, lattices, ordered algebraic structures Definition

Krull dimension of a poset

Definition

Let $(P, \le)$ be a partially ordered set. A finite chain $x_0 < x_1 < \cdots < x_n$ in $P$ has length $n$ . The Krull dimension $\dim P$ of a nonempty $P$ is the supremum of the lengths of the finite chains in $P$ , an element of $\N \cup \set{\infty}$ . The empty poset has Krull dimension $-\infty$ by convention; some authors use $-1$ instead.

Remarks

Example: the power set of $\set{1, \ldots, n}$ ordered by inclusion has Krull dimension $n$ . The chain $\varnothing \subset \set{1} \subset \cdots \subset \set{1, \ldots, n}$ has length $n$ , and a longer chain would need two subsets of the same size. Example: $\N$ with its usual order has Krull dimension $\infty$ . The dimension can be infinite even when every chain is finite: the disjoint union of chains of lengths $1, 2, 3, \ldots$ has no infinite chain but has dimension $\infty$ . For a commutative ring $R$ , the Krull dimension of the ring is the Krull dimension of its set of prime ideals ordered by inclusion.

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Chain concept
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Krull dimension of a poset definition

Tag	`STSV9XY3R5`
URI	`m2://STSV9XY3R5`
Kind	definition
Writing	W0 stub
MSC	06A06, 13C15
Defines	Krull dimension of a poset
Uses	Partial order, Chain
Changed	2 November 2026

Krull dimension of a poset

Definition

Remarks

Reading path

Further reading

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