$\overline{Y}$ is irreducible in $A:=\Bbb{C}[X,Y]/ (X^2-Y^3)$. I would like to prove that $A:=\Bbb{C}[X,Y]/ (X^2-Y^3)$ is not a UFD. This is equivalent to find an irreducible element which is not prime. 
I can prove that every element looks like $\overline{P_1X+P_0}$ for $P_1,P_0\in \Bbb{C}[Y]$. After that let $\overline{Y}$ be the "image" of $Y$ in the quotient. It's not prime because the ring $A/(\overline{Y})$ will be isomorphic to $\Bbb{C}[X]\ (X^2)$. 

But I am stuck to prove that $Y$ is irreducible, 

that is, I write: $$Y=P_1Q_1X^2+P_0Q_1X+P_1Q_0X+P_0Q_0
=P_1Q_1Y^3+P_0Q_0+(P_0Q_1+P_1Q_0)X$$
So by unicity on $\Bbb{C}(Y)[X]$, it's equivalent to 
$$Y=P_1Q_1Y^3+P_0Q_0$$ and
$$(P_0Q_1+P_1Q_0)X=0.$$
Now taking the value at zero in the first equation, we get that $P_0(0)=0$ (by symmetry). It means that $Y$ divides $P_0$. 
Question: How can we continue ?
 A: Let me denote by $x,y$ the residue classes of $X,Y$ modulo the ideal $(X^2-Y^3)$. Suppose $y=(ax+b)(cx+d)$. Then $Y=(aX+b)(cX+d)+(X^2-Y^3)e$ with $a,b,c,d\in\mathbb C[Y]$, $e\in\mathbb C[X,Y]$. Now send $X$ to $T^3$, $Y$ to $T^2$ and get $$T^2=[a(T^2)T^3+b(T^2)][c(T^2)T^3+d(T^2)].$$ This writes $$T^2=a(T^2)c(T^2)T^6+[a(T^2)d(T^2)+b(T^2)c(T^2)]T^3+b(T^2)d(T^2).$$ For $T\mapsto 0$ we get $b(0)d(0)=0$, so $b(0)=0$ or $d(0)=0$. Suppose $b(0)=0$ and then $b(Y)=Yb_1(Y)$. We now get $$1=a(T^2)c(T^2)T^4+[a(T^2)d(T^2)+b(T^2)c(T^2)]T+b_1(T^2)d(T^2).$$ For (odd) degree reasons $ad+bc=0$. It follows that $$1=a(T^2)c(T^2)T^4+b_1(T^2)d(T^2).$$ Then we must have $\deg a+\deg c+2=\deg b_1+\deg d$. But from $ad+bc=0$ we have $\deg a+\deg d=1+\deg b_1+\deg c$, so $\deg a=-\deg d+1+\deg b_1+\deg c$ and then $-\deg d+1+\deg b_1+\deg c+\deg c+2=\deg b_1+\deg d$, that is, $2\deg c+3=2\deg d$, a contradiction.
A: The ring is isomorphic to $\mathbb{C}[T^2,T^3] \subseteq \mathbb{C}[T]$ via $\overline{X} \mapsto T^3$ and $\overline{Y} \mapsto T^2$. Here, $T^2$ is irreducible, since the only non-trivial factors in $\mathbb{C}[T]$ are $\lambda \cdot T$ (with $\lambda \in \mathbb{C}^{\times}$), which do not lie in $\mathbb{C}[T^2,T^3]$.
