kk=ZZ
R=kk[x1,x2,x3,x1',x2',x3',y0,y1,y2,y3, Weights=>{1,1,1,100,100,100,1,1,1,1}]
I=ideal(x1*x3*y0-(x2^3+x1^2+x3^2),y1*x1*x2*x3-(x1*x2^3+x2^3*x3+x1^3+x3^3),y2*x1^2*x2*x3^2-(x2^6+2*x1^2*x2^3+x1*x2^3*x3+2*x2^3*x3^2+x1^4+x1^3*x3+x1*x3^3+x3^4),y3*x1^3*x2^2*x3^3-(x2^9+3*x1^2*x2^6+3*x2^6*x3^2+3*x1^4*x2^3+3*x1^2*x2^3*x3^2+3*x2^3*x3^4+x1^6+2*x1^3*x3^3+x3^6),x1*x1'-(x2^3+x3^2),x2*x2'-(x1^3+x3^3),x3*x3'-(x1^2+x2^3))
J=saturate(I,x1*x2*x3)
S=R/J
D=ideal(x1*x2*x3,x1'*x2*x3,x1*x2'*x3,x1*x2*x3')
D'=ideal(x1*x2*x3)
M=saturate(D',D)
D'==M