A Noetherian local ring (R, m) whose Krull dimension equals the minimal number of generators of its maximal ideal m; equivalently R has finite global (homological) dimension equal to that Krull dimension.

Principle

Regularity is organized by the equality between intrinsic geometric dimension (Krull dimension) and embedding dimension (the minimal number of generators of m); this equality encodes homological finiteness and the absence of certain relations among generators.

Demonstration

The formal power series ring k[[x1,...,xn]] or the localization k[x1,...,xn]_{(x1,...,xn)} are regular local rings of dimension n; their maximal ideal is generated by the coordinate parameters and global homological dimension equals n.

Misapplication

Declaring a ring regular because its maximal ideal can be generated by n elements without verifying Noetherian hypotheses or equality with Krull dimension; confusing a regular element or a regular sequence with the ring being a regular local ring.

Consequence

Modules over a regular local ring have finite projective dimensions bounded by the ring dimension; regular local rings are Cohen–Macaulay and their spectra are smooth at the closed point, enabling predictable deformation and intersection behavior.

Reversal

An irregular (singular) local ring has embedding dimension strictly larger than its Krull dimension; such rings exhibit homological pathologies (infinite projective dimensions for some modules) and are local models of singularities.

Boundary

Applies only to Noetherian local rings; it excludes non-Noetherian rings, non-local rings (unless applied to each localization at maximal ideals), and statements about global smoothness of non-affine schemes without local verification.

Semantic Tension

The term 'regular' also denotes smoothness in geometry and nonvanishing conditions in linear algebra; in this algebraic sense 'regular' is a precise homological/geometric equality rather than a generic notion of 'well-behaved'.

Synthesis

A regular local ring is a Noetherian local algebraic model whose dimension equals the minimal number of local parameters, equivalently characterized by finite homological dimension; it is the algebraic avatar of a nonsingular point.