Ring isomorphisms, question about ideals. Let $R$ be a commutative ring and let $I$ and $J$ be two ideals of the ring. Define the function $\phi:R\to R/I \times R/J$ by $\phi(x)=(x+I,x+J)$ is it then true that for $R/\ker(\phi) = R/(I \cap J)$ we have: $$R/(I \cap J) \cong R/I \times R/J$$ If so, why? Because I cant see this.

Only if I and J are coprime. This is a special case of the Chinese Remainder Theorem. The proof should be contained in any introductionary book concerning ring theory.
