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Show that if $I + J = R$, then $R/(I \cap J) \cong R/I \times R/J$

This is on a problem sheet for my class. I haven't a clue where to begin, this is the first isomorphism with a product that I've seen. All I've found so far is that there is some $a$ from $I$ and $b$ from $J$ such that $a + b = 1$ since $1$ is an element of $R = I + J$, and $I$ don't see how that helps.

user26857
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user135732
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  • First try to prove, for reltively prime pair of integers, $m,n$, that $\mathbf{Z}/mn\mathbf{Z}\to \mathbf{Z}/m\mathbf{Z}\times \mathbf{Z}/n\mathbf{Z}$ is an isomorphism (use Bezout's theorem on gcd) – P Vanchinathan Mar 16 '14 at 04:26
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    That's the next problem in the sequence, we're supposed use our solution from this problem to solve that problem rather than the other way around – user135732 Mar 16 '14 at 04:33

1 Answers1

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For any $r,s\in R$, we can write $r=r_i+r_j$ and $s=s_i+s_j$, where $r_i,s_i\in I$, etc. Let $a=r_i+s_j$, so $a-r=s_j-r_j\in J$, and $a-s=r_i-s_i\in I$. In particular, $a+I=s+I$, and $a+J=r+J$. This shows that the map $$ \varphi\colon R\to R/I\times R/J:a\mapsto (a+I,a+J) $$ is surjective. Verify it is a homomorphism, find its kernel, and apply the isomorphism theorems to get the result.

Ben West
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