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Commutative rings with unit must have a maximal ideal by Krull's theorem.

But is it true, in general, that such sets must have a unique maximal ideal?

Does it matter if the ring is finite or infinite?

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    Is it called Krull's theorem? And for your other two questions: Consider the integers, and Z/6.2012-11-12
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    Answered my own question..... If we have $C([0,1])$ then there are an infinite number of maximal ideals; for any $x \in [0,1]$ define $M_x = \{f|f(x)=0\}$ and this can be shown to be a maximal ideal.2012-11-12
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    Commutative unital rings with a unique maximal ideal form a special class of rings, called "local rings". It is possible to have commutative rings with one, or any finite number, or an infinite number of maximal ideals.2012-11-12
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    @DylanWilson: I think what the OP is calling Krull's theorem is the statement that any ideal maximal with respect to not intersecting with a multiplicative subset of a ring is prime. I also believe Matsumara uses this terminology, though I could be wrong.2012-11-12

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