Isomorphism of quotient rings with Cartesian product

Question

If $I$ and $J$ are ideals in $R$ and $S$ respectively, how do I show that $(R\times S)/(I\times J)\cong(R/I)\times(S/J)$? I started by showing that $I\times J$ is an ideal in $R\times S$ but am unsure how to proceed from here. Is the First Isomorphism Theorem perhaps useful here?

Answer 1

Hint: Show that $I \times J$ is the kernel of the surjection $R \times S \to (R/I) \times (S/J)$. Then use the first isomorphism theorem.

Answer 2

Consider the canonical surjective homomorphism (epimorphism): $R\rightarrow R/I:r\mapsto r+I$ with kernel $I$.

This gives rise to the epimorphism $\phi:R\times S\rightarrow R/I\times S/J: (r,s)\mapsto (r+I,s+J)$.

The kernel is $I\times J$ and so by the Homomophism theorem:

$(R\times S)/\ker(\phi)$ is isomorphic to $im(\phi)= R/I\times S/J$.