The question is as follows:
Show that the $\mathbb{C}$-algebras: $A=\mathbb{C}[x,y]/(x^2y-xy)$, $B=\mathbb{C}[x,y]/(x^2y+xy^2)$, $C=\mathbb{C}[x,y,z]/(xy, yz, zx)$, and $D=\mathbb{C}[x,y]/(x^2y+xy^2+x^4+y^4)$ are pairwise non-isomorphic.
Is there a particularly elegant way to approach this? I've tried playing around with decompositions, but I can't quite get things to work. Any help would be much appreciated! Thanks in advance.
STRONG HINTS
Look at how the algebraic sets decompose. The variety corresponding to $A$ is the union of three lines intersection in two points. It looks like this $\rightarrow \coprod$, because $(x^2y-xy)=(x) \cap (y) \cap(x-1)$.
The second decomposes as $(x) \cap (x+y) \cap (y)$, so the algebraic set again consists of three lines, but this time they intersect in three points, forming a "fan" of three lines.
Similarly, $C$ is the union of the coordinate axes, so they intersect in one point.
And the fourth equation is irreducible, so $D$ is an integral domain.