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I've been asked to show that in a right Noetherian domain, the intersection of nonzero right ideals is nonzero. A hint is given, saying that if not, then any nonzero right ideal contains a direct sum of two nonzero right ideals. Assuming this, I believe that I can argue as follows:

Let $I$ be a right ideal. Then $I$ contains a direct sum of nonzero right ideals, $I \supseteq I_1 \oplus I_2$. We can then repeat this process on $I_1$ (or $I_2$) to obtain an infinite direct sum of nonzero right ideals contained in $I$, $I \supseteq \bigoplus_{i \in \mathbb{N}} I_i$. But every right ideal contained in $I$ should be finitely generated, and $\bigoplus_{i \in \mathbb{N}} I_i$ cannot possibly be finitely generated.

However, I can't see how to prove the result in the hint. I can see that if we have $I \cap J = 0$, say, then $I \oplus J$ is direct and then for any right ideal $K$ we have $(I \cap K) \oplus (J \cap K) \subseteq K$, but I can't see why these summands should be nonzero.

Any hints on how to proceed?

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  • $\begingroup$ It should've been right ideals throughout. $\endgroup$ – FrancisW Nov 4 '14 at 9:10
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Let $k\in K$ be nonzero. Look at $kI+ kJ\subseteq K$.

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  • $\begingroup$ Yep, that does the trick. Thanks for your help. $\endgroup$ – FrancisW Nov 4 '14 at 9:10
  • $\begingroup$ @FrancisW no problem :) $\endgroup$ – rschwieb Nov 4 '14 at 10:50

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