Regular ring (in commutative algebra)

A Noetherian ring whose localizations (cf. Localization in a commutative algebra) are all regular (here is a prime ideal in ). A local Noetherian ring (cf. Local ring) with maximal ideal is called regular if is generated by elements, where , that is, if the tangent space (as a vector space over the field of residues) has dimension equal to . This is equivalent to the absence of singularities in the scheme . A regular local ring is always integral and normal, and also factorial (cf. Factorial ring; the Auslander–Buchsbaum theorem), and its depth is equal to (cf. Depth of a module). The associated graded ring

is isomorphic to the polynomial ring . A local Noetherian ring is regular if and only if its completion is regular; in general, if is a flat extension of local rings and is regular, then is also regular. For complete regular local rings, the Cohen structure theorem holds: Such a ring has the form , where 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 theorem on syzygies); the converse also holds (see [2]).

Fields and Dedekind rings are regular rings. If is regular, then the ring of polynomials and the ring of formal power series over are also regular. If is a non-invertible element of a local regular ring, then is regular if and only if .

References

[1]	O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975) MR0389876 MR0384768 Zbl 0313.13001
[2]	J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965) MR0201468 Zbl 0142.28603
[3]	A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique. I. Le langage des schémas" Publ. Math. IHES , 4 (1964) MR0173675 Zbl 0118.36206