In Mumford's "Red Book of Varieties and Schemes", Chapter III, Paragraph 10 (entitled "Flat and smooth morphisms"), the following property is stated:
Let $M$ be a $B$-module, and $B$ an algebra over $A$. Let $f\in B$ have the property that for all maximal ideals $m \subset A$, multiplication by $f$ is injective in $M / m \cdot M$. Then $M$ flat over $A$ implies $M / f \cdot M$ flat over $A$.
My question is whether this statement is true as stated, without some finiteness assumptions (it is not clear, for example, what is the role of $B$ here if no finiteness is assumed), and, secondly, can someone indicate a proof, or an exact reference for a proof.
Thank you, Sasha