# Regular ring (in commutative algebra)

A Noetherian ring $A$ whose localizations (cf. Localization in a commutative algebra) $A _ {\mathfrak p }$ are all regular (here $\mathfrak p$ is a prime ideal in $A$). A local Noetherian ring $A$( cf. Local ring) with maximal ideal $\mathfrak m$ is called regular if $\mathfrak m$ is generated by $n$ elements, where $n = \mathop{\rm dim} A$, that is, if the tangent space $\mathfrak m / \mathfrak m ^ {2}$( as a vector space over the field of residues) has dimension equal to $\mathop{\rm dim} A$. This is equivalent to the absence of singularities in the scheme $\mathop{\rm Spec} A$. A regular local ring $A$ is always integral and normal, and also factorial (cf. Factorial ring; the Auslander–Buchsbaum theorem), and its depth is equal to $\mathop{\rm dim} A$( cf. Depth of a module). The associated graded ring
$$G _ {\mathfrak m } ( A) = \ \oplus _ {i \geq 0 } \mathfrak m ^ {i} / \mathfrak m ^ {i+} 1$$
is isomorphic to the polynomial ring $k [ X _ {1} \dots X _ {n} ]$. A local Noetherian ring $A$ is regular if and only if its completion $\widehat{A}$ is regular; in general, if $A \subset B$ is a flat extension of local rings and $B$ is regular, then $A$ is also regular. For complete regular local rings, the Cohen structure theorem holds: Such a ring has the form $R [ [ X _ {1} \dots X _ {n} ] ]$, where $R$ is a field or a discrete valuation ring. Any module of finite type over a regular local ring has a finite free resolution (see Hilbert syzygy theorem); the converse also holds (see ).
Fields and Dedekind rings are regular rings. If $A$ is regular, then the ring of polynomials $A [ X _ {1} \dots X _ {n} ]$ and the ring of formal power series $A [ [ X _ {1} \dots X _ {n} ] ]$ over $A$ are also regular. If $a \in A$ is a non-invertible element of a local regular ring, then $A / aA$ is regular if and only if $a \notin m ^ {2}$.