I'm stuck on the following exercise from Herstein's "Topics in Algebra":
"show that ($n$ is prime) $\Leftrightarrow ([a][b]=\Rightarrow [a]=[b]=) $ in $J_n$".
for the rightward implication I have:
$[a][b]=[ab]=\Rightarrow n|ab\Rightarrow n|a$ or $n|b$ (or both) $\Rightarrow [a]=$ or $[b]=$
and I'm stuck on the leftward implication.
Why do I get $[a]=$ OR $[b]=$ and not $[a]=$ AND $[b]=$ in the leftward implication? Where is it that I go wrong?
I'd also appreciate any comment/hint about how to prove the remaining left implication.