I'm currently reading about PIDs and have come across a question involving maximal ideals which at one point reads "Suppose that a Euclidean domain $R$ had a unique maxima ideal $P$". Does this mean that $P$ is the only maximal ideal? Or that if $P = \langle p \rangle$ it cannot also equal $\langle p_1 \rangle$ for another prime $p_1$?
2026-03-25 17:20:59.1774459259
What exactly does it mean for a maximal ideal to be unique in a principal ideal domain?
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Each nonzero ring with identity (even the noncommutative ones) has at least one maximal ideal. To say it has a unique maximal ideal means that it has exactly one and no more.
Commutative rings which have a unique maximal ideal are called local rings, and they are a key thing in commutative algebra.