I need to find a sufficient prerequisite on formal language $L$ over alphabet $\Sigma$ so that $L^*$ is a finite language.
I say that language $L^*$ is finite if and only if $L = \{ \varepsilon \}$, because if $\exists w\in L, |w|>0$ then $w^i\in L^*$ for $i>0$ and for all $i \neq j$, $w^i \neq w^j$, therefore $L^*$ is infinite.
But as this question came up after learning Nerode theorem, I was intrigued if a better prerequisite can be chosen. As every finite language is regular, from Nerode theorem I get that $rank(R_{L^*})$ is finite which means that $rank(R_{L})$ is also finite, so can I say that if $rank(R_{L})$ is finite then $L^*$ is finite? Or would it only mean that $L^*$ is regular but not necessarily finite