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I have a solution to this question which I know to be wrong, but I can't find the mistake! The question reads "Let $A$ be a ring in which every element $x$ satisfies $x^n = x$ for some $n>1$. Show that every prime ideal in $A$ is maximal."

My solution goes:

We have $x^n-x = 0 \implies x(x^{n-1}-1) = 0$ so either $x$ is nilpotent or a unit or 0. If $x$ is nilpotent then so is $x^{n-1}$ as the nilradical is an ideal and so by Question 1 we have $x^{n-1}-1$ is a unit in $A$. Thus $$0 = x(x^{n-1}-1)(x^{n-1}-1)^{-1} = x.$$ Hence every non-zero element of $A$ is a unit and then the only prime and maximal ideal is (0).

I can't see where I've made the mistake but I must have somewhere otherwise it implies that all Boolean rings have two elements.

Pacifism
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    $A$ is not a domain of integrity, so $x(x^{n-1}-1)=0$ does not imply necessarily that $x$ is zero, a unit or nihilpotent, as both the factors could be different from $0$. –  May 17 '20 at 11:49
  • Hint $ $ Reduce to the domain case by factoring out by a prime. See the Linked questions list there for many more examples of this basic method. – Bill Dubuque May 17 '20 at 17:34

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We have $x^n-x = 0 \implies x(x^{n-1}-1) = 0$ so either $x$ is nilpotent or a unit or 0.

There are two mistakes in one here. It appears you think that you are able to conclude that $x=0$ or $x^{n-1}-1=0$ from this line.

Firstly, as mentioned in a comment, this does not have to be an integral domain, and it need not be true.

Secondly, $x^{n-1}-1=0$ does not imply $x$ is nilpotent, it implies $x$ is a unit.

But basically the reasoning you gave is relevant, because you should be applying all this reasoning to $R/P$ where $P$ is a prime ideal. That proves that $R/P$ is a field, therefore $P$ is maximal.

rschwieb
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    The duplicate-hub for the main question seems to be this, although since you successfully phrased your question as a "I know this is wrong but can't see the mistake" question, I'm inclined not to classify it as a duplicate. But just in case, make sure you search thoroughly first, in the future. Reading the duplicate solutions might have made you realize your mistake, and that would have been a better learning outcome for you. – rschwieb May 17 '20 at 12:38