Examples
- The dimension of a polynomial ring over a field is the number of indeterminates d. These rings correspond to affine spaces in the language of schemes, so this result can be thought of as foundational. In general, if R is a Noetherian ring of dimension d, then the dimension of R is d + 1. If the Noetherianity hypothesis is dropped, then R can have dimension anywhere between d + 1 and 2d + 1.
- The ring of integers has dimension 1. More generally, any principal ideal domain that is not a field has dimension 1.
- An integral domain is a field if and only if its Krull dimension is zero. Dedekind domains that are not fields (for example, discrete valuation rings) have dimension one. In general, a Noetherian ring is Artinian if and only if its Krull dimension is 0.
- An integral extension of a ring has the same dimension as the ring does since it has "incompatibility" and "going up".
- A noetherian local ring is called a Cohen–Macaulay ring if its dimension is equal to its depth. A regular local ring is an example of such a ring.
Read more about this topic: Krull Dimension
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