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

Question

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

Answer 1

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.