### Testing the Gorenstein and regular property for Stanley-Reisner rings using Sage

One can obtain in Sage the Stanley-Reisner ring of a simplicial complex as follows:

X = SimplicialComplex([[0,1,2], ~~[0,2,3]])
~~[0,2,3]])

X.stanley_reisner_ring()

There is also a given method to check whether the Stanley-Reisner ring is Cohen-Macaulay using Sage as follows:
X.is_cohen_macaulay(QQ)

A natural question in commutative algebra is whether a Cohen-Macaulay ring is a Gorenstein ring or even a regular ring. Is there a method to check this for a given commutative ring with Sage or at least for Stanley-Reisner rings?