Commutative Ring - Dimension

Dimension

The Krull dimension (or simply dimension) dim R of a ring R is a notion to measure the "size" of a ring, very roughly by the counting independent elements in R. Precisely, it is defined as the supremum of lengths n of chains of prime ideals

0 ⊆ p₀ ⊆ p₁ ⊆ ... ⊆ p_n.

For example, a field is zero-dimensional, since the only prime ideal is the zero ideal. It is also known that a commutative ring is Artinian if and only if it is Noetherian and zero-dimensional (i.e., all its prime ideals are maximal). The integers are one-dimensional: any chain of prime ideals is of the form

0 = p₀ ⊆ pZ = p₁, where p is a prime number

since any ideal in Z is principal.

The dimension behaves well if the rings in question are Noetherian: the expected equality

dim R = dim R + 1

holds in this case (in general, one has only dim R + 1 ≤ dim R ≤ 2 · dim R + 1). Furthermore, since the dimension depends only on one maximal chain, the dimension of R is the supremum of all dimensions of its localisations R_p, where p is an arbitrary prime ideal. Intuitively, the dimension of R is a local property of the spectrum of R. Therefore, the dimension is often considered for local rings only, also since general Noetherian rings may still be infinite, despite all their localisations being finite-dimensional.

Determining the dimension of, say,

k / (f₁, f₂, ..., f_m), where k is a field and the f_i are some polynomials in n variables,

is generally not easy. For R Noetherian, the dimension of R / I is, by Krull's principal ideal theorem, at least dim R − n, if I is generated by n elements. If the dimension does drops as much as possible, i.e. dim R / I = dim R − n, the R / I is called a complete intersection.

A local ring R, i.e. one with only one maximal ideal m, is called regular, if the (Krull) dimension of R equals the dimension (as a vector space over the field R / m) of the cotangent space m / m2.