regular local ring

A **regular local ring** $R$ is a Noetherian commutative unital local ring whose Krull dimension? agrees with the minimal number of generators of its maximal ideal $I$. Equivalently, the Krull dimension equals $dim_k (I/I^2)$ where $k = R/I$ is the residue field of $R$. A Noetherian local ring is regular iff its global dimension is finite; it follows that its global and Krull dimension coincide.

Last revised on January 28, 2014 at 11:33:56. See the history of this page for a list of all contributions to it.