Non boolean example of a finite ring $R$ with $r^4 = r$ for all $r$ in $R$.

Question

I just proved that a finite ring $R$ with $r^4 = r$ for all $r$ in $R$ must be commutative.

But I don't see any non boolean example to ilustrate.

Answer 1

The field with four elements will do. Just consider $F_2$, the field with two elements and $F_2[X]/(x^2+x+1)$ will be your example.

Answer 2

Actually one can show that every finite ring satisfying this identity is a product of copies of $\mathbb{F}_2$ and $\mathbb{F}_4$. As you have said, it is commutative, and it is also reduced and of dimension $0$. Hence, it is a product of finite fields. But finite field $F$ has this property iff $|F^*|$ divides $4-1=3$, i.e. iff $F=\mathbb{F}_2$ or $F=\mathbb{F}_4$.