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.)

  • 2
    $\begingroup$ I think it's completely unnecessary to edit in a "thank you" message on behalf of the OP. $\endgroup$
    – Emily
    Commented Jul 26, 2013 at 15:54
  • $\begingroup$ What do you mean by an "indecomposable ideal"? $\endgroup$
    – user26857
    Commented Jul 26, 2013 at 15:55
  • $\begingroup$ An ideal such that it doesn't possess a primary decomposition. $\endgroup$ Commented Jul 26, 2013 at 16:12
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    $\begingroup$ See my answer here. (Btw, you can add the definition of indecomposable ideals into the body of your question.) $\endgroup$
    – user26857
    Commented Jul 26, 2013 at 16:34


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