Using elementary ring theory to show that $\langle xy^3, x^2y^2, x^3y\rangle$ cannot be generated by 2 elements over $\mathbb{C}[x,y]$

Question

I am trying to show that the ideal $\langle xy^3, x^2y^2, x^3y\rangle$ in $\mathbb{C}[x, y]$ cannot be generated by 2 elements.

I found a proof online but it was quite involved. This question is at the end of an introductory text on rings. Does anyone have an idea of how to do this?

Answer 1

The quotient I/J of that ideal by the ideal J generated by all monomials of degree 5 has dimension 3. Show that if we had two polynomials f and g which generate I, the images of f and g in I/J would generate the quotient I/J. That is absurd.