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If two prime ideals contain the same non trivial idempotents, what can we say about those ideals? Are they equal?

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Is there any reason you think this is true? Some context would help people answer, and probably help give answers that are better for your purpose –  Joe Tait Feb 24 at 9:13
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Consider the ring $\mathbb{Z}\times\mathbb{Z}$ (which is arguably the simplest ring having non-trivial idempotents, so it should be one of the first examples we look at). Its non-trivial idempotents are $(1,0)$ and $(0,1)$, and its prime ideals are those ideals of the form $\mathbb{Z}\times P$ and $P\times\mathbb{Z}$ where $P\subset\mathbb{Z}$ is a prime ideal. Thus $\mathbb{Z}\times (2)$ and $\mathbb{Z}\times (3)$ are both prime ideals containing the same non-trivial idempotents (namely, only $(1,0)$) while not being identical.

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Since the OP didn't actually require the ring to have any nontrivial idempotents, even $\mathbb Z$ would give a counterexample, –  Andreas Blass Feb 24 at 14:48
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