Show that $(A \times B) \big/ (\mathfrak{a} \times \mathfrak{b}) \cong \big(A \big/ \mathfrak{a} \big) \times \big( B \big/ \mathfrak{b} \big)$. Where $A$ and $B$ are commutative rings.
I am asked to prove this question but I am stuck. Here are my thoughts:
Consider the map $f: A \times B \longrightarrow (A/\mathfrak{a}) \times (B/\mathfrak{b})$ given by $(x,y) \longrightarrow (\bar{x},\bar{y})$.
I believe that I should use the First Isomorphism Theorem for rings to show that the kernel of $f$ is $\mathfrak{a} \times \mathfrak{b}$ which is an ideal of $A \times B$. Also I believe I need to show that the map $f$ is a ring homomorphism.
Does any of this sound reasonable? If so, where would I go from here? If not, where should I begin?
Thank you for your help!!!
Yes, that is a good idea. Show that $f$ is a surjective homomorphism. Then note that $(x,y) \in \ker f$ iff $x \in \mathfrak{a}$ and $y \in \mathfrak{b}$ iff $(x,y) \in \mathfrak{a} \times \mathfrak{b}$.