To put this into context, I was doing a reading on the text with my mentor and the following step proved to be unjustifiable to the both of us.
The step I am referring to is the paragraph that goes "Similarly, the module $N\otimes M$ is generated by the elements ..."
In particular, I am confused on how Eisenbud is able to justify that $N$ is finitely generated just because the relations of bilinearity of $N\otimes M$ involves finitely many elements.
Besides this step, the rest of the proof makes sense to me. Any ideas?
