# Are the two prime ideals containing same idempotents always the same?

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
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.
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