Intereting Posts

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in normed space hyperplane is closed iff functional associated with it is continuous
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Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group

Let

Rbe a commutative ring with an identity such that for all $r\in$R, there exists some $n>1$ such that $r^n = r$. Show that any prime ideal is maximal. (Atiyah and MacDonald,Introduction to Commutative Algebra, Chapter 1, Exercise 7.)

Any hints?

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Hint: Reduce to the case that $R$ is an integral domain satisfying $\forall r \exists n (r^n=r)$, and show that $R$ is a field.

Let $\,D\,$ be any integer domain and let $\,d\in D\,$ be s.t. $\,d^n=d\,\,,\,\,1<n\in\Bbb N\,$ , then:

$$d^n=d\Longrightarrow d(d^{n-1}-1)=0\Longleftrightarrow d=0\,\,\,\text{or}\,\,\,d^{n-1}=1,$$

so if $\,d\,$ is not zero then it *must* be a unit.

$$——-o———-o———o—$$

In our case: let $\,I\leq R\,$ be a prime ideal and let $\, r\in R\setminus I\,$, then:

$$\exists\,n\in\Bbb N\,\,s.t.\,\,r^n=r\Longrightarrow \left(r+I\right)^n=r^n+I=r+I\in R/I$$

Now use the first part with $\,D:=R/I\,\,\,,\,\,d=r+I\,$ and deduce $\,R/I\,$ is actually a field…

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