In Eisenbud's book Commutative Algebra with a View Toward Algebraic Geometry, in the prove of Proposition 1.4, the auther seems to use the following fact.
Let $R$ be a Noetherian ring, $M$ is a finitely generated $R$-module, $N$ is a submodule of $M$, and $f_1\in M$. Then the author says:
Since $Rf_1\subset M$ is generated by one element, its submodule $N\cap Rf_1$ is finitely generated.
My question is, I don't think that if $K$ is finitely generated (even with one element), then any submodule of $K$ is also finitely generated. For example, any Non-Noetherian ring $R$ is finitely generated by ONE element, say 1, but there exists an ideal which is not finitely generated.
Why can Eisenbud deduce such conclusion?