TITLE: Finding ring isomorphisms
QUESTION [0 upvotes]: Let $A$ be a ring with $0\neq 1$ such that $x^4=1, \forall x\in A$, with $x\neq 0$. My question is: to which ring is $A$ isomorphic? 

$A$ can be, for example, isomorphic to $\mathbb{Z}_2$. The question arose when I proved that a ring with $0\neq 1$ and $x^2=1,\forall x\in A $, with $x\neq 0$ is isomorphic to $\mathbb{Z_2}$ or $\mathbb{Z_3} $. Thank you!

REPLY [1 votes]: Note the following theorem of Jacobson (see, e.g., T.Y. Lam, A First Course in Noncommutative Rings, Theorem 12.10).
Theorem Let $A$ be a ring such that, for any $x \in A$ there exists an integer $n(x) > 1$ such that $x^{n(x)}=x$. Then $A$ is commutative.
Now let $A$ be a ring satisfying your property. Since $x^5=x$ for all $x 
\in A$, the ring is commutative by the theorem of Jacobson. Since $x^4=1$ for all $x \in A \setminus \{0\}$, every nonzero element is invertible. Hence $A$ is a field. All nonzero elements of $A$ are roots of the polynomial $T^4-1$. Thus $A$ has at most five elements.
So $A$ must be isomorphic to one of $\mathbb F_2$, $\mathbb F_3$, $\mathbb F_4$, or $\mathbb F_5$. In $\mathbb F_4$, we have $x^4=x$ for all $x \in \mathbb F_4$, and hence your property is not satisfied. However, for the other fields it is satisfied, and it follows that your property holds if and only if $A$ is isomorphic to one of $\mathbb F_2$, $\mathbb F_3$, or $\mathbb F_5$.
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