Local Rings, Regular Ring
Regular Ring
A local noetherian commutative ring is regular if the dimension of R equals the dimension
of M/M2 as a K vector space, where K is the residue field R/M.
For instance a discrete valuation ring is regular.
Its dimension is 1, with M the only nonzero prime ideal,
and M is principal, generated by some t, so the image of t mod M2 generates
the K vector space.
Since M > M2,
t does not lie in M2, and the vector space has rank 1.
If R has dimension 0 it is artinian, as shown in the last section.
If the maximal ideal M is 0, then M/M2 = 0, and R is regular.
If M is nonzero then M2 is properly contained in M, since M is nilpotent.
This gives a positive dimension, and R is not regular.
The only regular rings of dimension 0 are fields.