module-finite

Let $S$ be a ring with subring $R$.

We say that $S$ is module-finite over $R$ if $S$ is finitely generated as an $R$-module.

We say that $S$ is ring-finite over $R$ if $S=R[v_1,\mathrm{\dots },v_n]$ for some $v_1,\mathrm{\dots },v_n\in S$.

Note that module-finite implies ring-finite, but the converse is false.

If $L$ is ring-finite over $K$, with $L,K$ fields, then $L$ is a finite extension of $K$.