Regular Local Ring - Examples

Examples

1. Every field is a regular local ring. These have (Krull) dimension 0. In fact, the fields are exactly the regular local rings of dimension 0.
2. Any discrete valuation ring is a regular local ring of dimension 1 and the regular local rings of dimension 1 are exactly the discrete valuation rings. Specifically, if k is a field and X is an indeterminate, then the ring of formal power series k] is a regular local ring having (Krull) dimension 1.
3. If p is an ordinary prime number, the ring of p-adic integers is an example of a discrete valuation ring, and consequently a regular local ring, which does not contain a field.
4. More generally, if k is a field and X₁, X₂, ..., X_d are indeterminates, then the ring of formal power series k] is a regular local ring having (Krull) dimension d.
5. If A is a regular ring, then it follows that the polynomial ring A and the formal power series ring A] are both regular.
6. If Z is the ring of integers and X is an indeterminate, the ring Z_{(2, X)} is an example of a 2-dimensional regular local ring which does not contain a field.