toFreeAlgebraQuotient -- Converts a Ring to a noncommutative ring

i1 : R = QQ[a,b,c,d, SkewCommutative=>{2,3}]

o1 = R

o1 : PolynomialRing, 2 skew commutative variable(s)

i2 : I = ideal(a*d-b*c)

o2 = ideal(- b*c + a*d)

o2 : Ideal of R

i3 : S = R/I

o3 = S

o3 : QuotientRing

i4 : S' = toFreeAlgebraQuotient(S)
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o4 = S'

o4 : FreeAlgebraQuotient

i5 : ideal S'

                                                                             
o5 = ideal (a*b - b*a, a*c - c*a, b*c - c*b, a*d - d*a, b*d - d*b, c*d + d*c,
     ------------------------------------------------------------------------
      2   2
     c , d , a*d - b*c)

o5 : Ideal of QQ <|a, b, c, d|>