A classical example of a ring $R$ with an indecomposable ideal is the ring $C(X)$ of real valued continuous functions on $X$, where the $(0)$ ideal is not decomposable. Does anyone know other examples of rings with a non decomposable ideal? (In particular, I'm looking for a non reduced one.)
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2I think it's completely unnecessary to edit in a "thank you" message on behalf of the OP. – Emily Jul 26 '13 at 15:54
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What do you mean by an "indecomposable ideal"? – Jul 26 '13 at 15:55
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An ideal such that it doesn't possess a primary decomposition. – Sabino Di Trani Jul 26 '13 at 16:12
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2See my answer here. (Btw, you can add the definition of indecomposable ideals into the body of your question.) – Jul 26 '13 at 16:34