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.

If you have a ring homomorphism $\phi:R\to S$, the induced morphism $R/\ker\phi\to S$ has the same image as $\phi$. Since the induced morphism is injective, saying it is an isomorphism is equivalent to saying that $\phi$ is surjective.
Note that I'm using “isomorphism” with the meaning “bijective homomorphism”; some authors (Herstein, for example) use “isomorphism” with the meaning “injective homomorphism”; so they say that two rings (groups, semigroups, …) are isomorphic when there exists an isomorphism of one onto the other.
