nLab regular local ring

Redirected from "regular ring".

Not be confused with von Neumann regular rings in noncommutative algebra.

Contents

Idea

For a local ring to be regular is a kind of a smoothness condition.

Definition

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 whose 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.

A Noetherian commutative ring RR (not necessarily local) is regular if for all prime ideals pRp\subset R the localization R pR_p is a regular local ring.

Properties

Relation to other classes

Every regular local ring is a complete intersection ring and a fortiori a Cohen-Macaulay ring.

A useful noncommutative analogue of a regular Noetherian ring is the notion of Artin-Schelter regular ring.

References

Last revised on June 25, 2024 at 15:35:41. See the history of this page for a list of all contributions to it.