nLab regular local ring

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

Idea

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

Definition

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 whose 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.

A Noetherian commutative ring $R$ (not necessarily local) is regular if for all prime ideals $p\subset R$ the localization $R_p$ 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

• Wikipedia, Regular local ring

• eom: Regular ring (in commutative algebra)