nLab
regular local ring

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

Revised on January 28, 2014 11:33:56 by Adeel Khan (77.9.213.189)