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 ).
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 .
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Regular ring (in commutative algebra). Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Regular_ring_(in_commutative_algebra)&oldid=24549