Commutative algebra: Irreducible components of $\operatorname{Spec}(A)$

Question

Let $k$ be a field of characteristic other than $2$. Consider the ring $$A:= k[X,Y]/(X(Y+1),X(Y+X^2)).$$ Describe all the irreducible components of $\operatorname{Spec}(A)$.

It does not look that hard, but I could not solve this on my exam on commutative algebra.

I know that if $\mathfrak{p}\in A$ is a minimal prime ideal, then $V(\mathfrak{p}) = \{\mathfrak{q} \text{ prime ideal } ∣ \mathfrak{p}\subseteq\mathfrak{q} \}$ is an irreducible component of $\operatorname{Spec}(A)$. But how can I use this on a quotient ring?

Answer 1

The minimal primes of $A$ correspond to the minimal primes over $I$. If $P$ is a minimal prime containing $I$, then $X(Y+1)\in P$, so either $X\in P$ or $Y+1\in P$. If $X\in P$, then $I\subseteq (X)\subseteq P$, so by minimality, $P=(X)$.

If $X\notin P$, then $Y+1, Y+X^2\in P$, so $I\subseteq (Y+1, Y+X^2)=(X^2-1, Y+1)\subseteq P$. Since $X^2-1\in P$, either $X-1\in P$ or $X+1\in P$, and once again by minimality, we conclude that either $P=(X-1,Y+1)$ or $P=(X+1, Y+1)$.