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
.