regular local ring
A regular local ring is the algebra of a smooth point — a place on a variety where there is no corner, cusp, crossing, or pinch. At a smooth point the space looks, infinitesimally, just like flat coordinate space, and the algebraic shadow of that smoothness is a precise efficiency: the maximal ideal needs no more generators than the dimension allows. Singular points 'waste' generators; regular points use exactly the minimum.
Let (R, m, k) be a Noetherian local ring with maximal ideal m and residue field k = R/m. Always one has dim R <= dim_k(m / m^2), the number of generators needed for m by Nakayama. R is regular when equality holds: dim R = dim_k(m / m^2), so m is generated by exactly d = dim R elements, a regular system of parameters. Equivalently the associated graded ring is a polynomial ring over k.
Regular local rings are remarkably good: by the Auslander–Buchsbaum–Serre theorem they are exactly the Noetherian local rings of finite global dimension, and a deep theorem (Auslander–Buchsbaum) says every regular local ring is a unique factorization domain. They are integrally closed and Cohen–Macaulay. The simplest examples are fields (d = 0), DVRs (d = 1), and power series rings k〚x_1, ..., x_d〛. The node k[x, y]/(y^2 - x^3) localized at the origin is the prototypical NON-regular local ring: dimension 1 but m needs 2 generators.
The local ring of k[x, y] at the origin (x, y) is regular: dimension 2, and m = (x, y) is generated by exactly 2 elements, with m/m^2 the 2-dimensional space spanned by x and y. The origin of the plane is a smooth point.
A smooth point: generators of m match the dimension.
Regularity localizes: if R is regular, so is every localization R_p. A variety over a perfect field is smooth at a point exactly when its local ring there is regular — this is the bridge between algebraic regularity and geometric smoothness.