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Let $R$ be a ring and $I$ and $J$ be ideals in $R$ such that $I + J = R$. Show that there exists a ring isomorphism $R/(I\cap J) \cong R/I \times R/J$. I've already proved that $IJ= I\cap J$. Feels like I am pretty closed to the answer, but trapped. Thanks!

R_D
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J.doe
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1 Answers1

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In truth, it would be more correct to use $R/I\oplus R/J$ instead of the product.

Let $r+I\cap J\in R/(I\cap J)$. Choose a decomposition $r = i+j$ with $i\in I$ and $j\in J$, and define a map $$\phi:R/(I\cap J)\longrightarrow R/I\oplus R/J$$ by $$\phi(r+I\cap J) = (j+I)\oplus(i+J).$$ You can easily show that this map is well defined, and it is a morphism of rings. It is also an easy exercise to show that it is bijective.

Daniel Robert-Nicoud
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