# 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 $S=R[v_{1},\ldots,v_{n}]$ 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 $K$.

Title module-finite Modulefinite 2013-03-22 12:36:56 2013-03-22 12:36:56 yark (2760) yark (2760) 6 yark (2760) Definition msc 13B02 msc 13C05 msc 16D10 FinitelyGeneratedRModule ring-finite