Suppose $p \subset q$ are two distinct prime ideals of the commutative ring $R.$ Let $K$ be the quotient field of $R/p.$ Show that $K$ is not finitely generated as $R$-module.
My guess: If $K$ is finitely generated as $R$-module it would be finitely generated as $R/p$-module, hence $K$ becomes an integral extension over $R/p.$ This implies that $R/p$ is a field which is not true, since $q/p$ is an nonzero prime ideal in $R/p.$ My only doubt is why $K$ can be considered as an $R$-module ? Is it obvious ?