On monomial ideals and ring generated by monomials

Question

Question 1: Is $(x^4,x^3y,x^2y^2,xy^3,y^4)$ a maximal ideal in $\mathbb C [ x^4,x^3y,x^2y^2,xy^3,y^4] $?

Question 2: Are the ideals $(x^4,x^3y,x^2y^2,xy^3,y^4)$ and $(x^4,x^3y,xy^3,y^4)$ distinct in the ring $\mathbb C[x^4,x^3y,x^2y^2,xy^3,y^4] $ ?

If the answer to both the questions is yes, then we get an example of ideals $I,J$ in a domain $R$ such that $I$ is maximal, $J\ne I$ and $I^2=J^2$.

Answer 1

Question 1: It is the kernel of the restriction of the surjective ring homomorphism $$\Bbb{C}[x,y]\ \longrightarrow\ \Bbb{C}:\ f(x,y)\ \longmapsto\ f(0,0),$$ to the ring $\Bbb{C}[x^4,x^3y,x^2y^2,xy^3,y^4]$, so it is maximal.

Question 2: Of course; the latter clearly does not contain $x^2y^2$.