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 $S=R[v_1,\ldots,v_n]$ for some $v_1,\ldots,v_n \in S$ .

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 $K$ .