$P + Q = R$ is clear because in PID prime ideals are maximal. However if induction is the key I don't know how to proceed. Please help.
2026-03-27 13:42:10.1774618930
Prove that for distinct prime ideals $P, Q$ of PID $R$, $P^n + Q^m = R$.
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If $P^n+Q^m$ is not equal to $R$,then exists a maximal ideal $I$ of $R$ such that $P^n+Q^m\subset I$.Hence $P\subset I$ and $Q\subset I$.Contradiction.