I would like to learn about sources that give a gentle introduction to flat modules, with exercises. Could you recommend me some source? This might be a section in a book, or some article.
I am reading the Primer in Commutative Algebra of James Milne, and I find the section on flatness harder than average in this (great) text. So I would like to find some other pedagogical source that would contain a bit more details and examples.
I like Osborne's Basic Homological Algebra very much. Check chapter 2 for a gentle introduction to proj. / inj. & flat modules. You can find some exercises at the end of it (check in particular ex. 11, aka the "flat test lemma").