Ring Homomorphism-fixed sets

Question

Given the homomorphism $f: \mathbb{Z}\times\mathbb{Z}\to \mathbb{Z} \times \mathbb{Z}$ given by $f (r,s) = (s,r)$. Find the fixed set.

Answer 1

Hint: if $g\colon X\to X$ is a map, the fixed points for $g$ are those for which $g(x)=x$.

For instance, if $g(x)=x^2$ (from the reals to the reals), we want $x^2=x$, that is, $x=0$ or $x=1$.

Now, in your case…