My guess is that if that condition holds then probably $P=R$ where $R$ is the ring .Am I on the right track?
2026-03-30 10:19:53.1774865993
Does $1 \in P$ where $P$ is a prime ideal.
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If $1\in P$ then for each $x\in R$ we have $x=x1\in P$. So yes, if any ideal contains the identity then it is the whole ring.