Let $A=k[x,y,z,a]/I$, where $$I= \lbrace a^2(x+1)-z^2, ax(x+1)-yz, xz-ay,y^2-x^2(x+1) \rbrace$$
I'd like to show that each class $a \in A$ is represented by a unique polynomial $f_a$ with the following property: no monomial term of $f_a$ is divisible by $z^2, yz, xz$, or $y^2$.
What is a good strategy for proving such a claim (in particular, the uniqueness)?