There is a nice result concerning flat modules over a ring:
If every finitely generated submodule of a module $M$ is flat, then $M$ is flat.
However, the proof I've read in Rotman's Homological Algebra is quite messy IMHO. In fact, it violates the golden rule of tensor products: never use its construction, use its universal property!!
I attach the proof to the end of the post. The "mess" is in the lemma.
So the question is:
- Is there a cleaner proof, also as elementary? (Rotman has just barely defined flat modules at that point, plus proven a few elementary properties)
(I'm thinking perhaps this proof can be translated to use the universal property...)
If you know a proof that is not as elementary, please do post it anyway, it should be interesting.
- If you think not, then could you give an heuristic* as to why you think there is not a cleaner proof? i.e., what is the fundamental reason (if there is any) of necessarily having to deal with the construction of the tensor product in the proof of this theorem?
*I'm not proficent in the usage of the word "heuristic", I apologize in advance if it's improperly used ;)