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You want to quotient by the two-sided ideal generated by x^2 + 1 and y^2:

R.<x,y> = P.quo(P*[x^2 + 1, y^2]*P)

Note this re-defines x and y once again, as you also did in your code.

Then you get:

sage: x^2
sage: y^2
sage: y*x