Let $Sec(\pi)$ denote the set of all (set-theoretic) cross sections
$Sec(\pi)=\{s:R/\sim \to R | \pi \circ s = 1_{R/\sim}\}$
Establish a bijective correspondence of normalization maps $R \to R$ with $Sec(\pi)$ and show that, under this bijection, homomorphisms correspond to homomorphisms.
So $Sec(\pi)$ is the set of all cross sections $s$ satisfying $\pi(s([x])=[x]$ for $[x] \in R/\sim$?
What is a normalization map? I'm stuck on this problem. Thanks for any help!