(F,G) = ringFromFractions(H,f)
Serious restriction: It is assumed that this ring R[1/f H] is an endomorphism ring of an ideal in $R$. This means that the Groebner basis, in a product order, will have lead terms all quadratic monomials in the new variables, together with other elements which are degree 0 or 1 in the new variables.
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The object ringFromFractions is a method function with options.