Krulldimensionen

Krulldimensionen, in English Krull dimension, is a numerical measure of the size of a commutative ring’s prime-ideal structure. For a ring R with unity, its Krull dimension is defined as the supremum of the lengths of all strictly increasing chains of prime ideals p0 ⊊ p1 ⊊ ... ⊊ pn within R, where the length is the number of inclusions. If no such bound exists, the Krull dimension is infinite. The dimension is denoted dim(R).

In practice, the Krull dimension captures a notion of complexity of a ring’s algebraic structure. It is always a nonnegative integer or infinity. For Noetherian rings, which satisfy the ascending chain condition on ideals, the Krull dimension is finite and can be read off from chains of prime ideals. Some standard examples: a field has Krull dimension 0; the ring of integers Z has Krull dimension 1, because chains like (0) ⊊ (p) have length 1; a polynomial ring in n variables over a field, k[x1, ..., xn], has Krull dimension n. More generally, for Noetherian rings R, dim(R[x1, ..., xn]) = dim(R) + n.

Other common properties include: dim(R/I) ≤ dim(R) for any ideal I, and dim(R × S) = max(dim(R), dim(S)). Local properties under localization, quotients, and products interact with the dimension in predictable ways, making Krull dimension a central concept in algebraic geometry and commutative algebra. In geometric terms, dim(R) equals the topological dimension of Spec(R), the spectrum of prime ideals, linking algebra to the dimension of algebraic varieties. The term Krullet dimensionen may refer to the dimension function across multiple rings in a mathematical context.