Bilinear Maps (Algebra 2: Lecture 19 Video 3)

Lecture 19: We started this lecture by discussing the free module on a set A.  We did not give a full proof of the main result here, but gave a summary of the ideas that go into it.  We then stated a corollary and some exercises that had consequences for the rank of a free module.  We then stated a result relating finitely generated R-modules and quotients of free modules.  We showed that Q is not a free Z-module.  We mentioned several exercises that develop some key properties and examples of R-modules.  In the last part of the lecture, as motivation for the material about tensor products coming up in the next few lectures, we discussed bilinear maps.  We gave a definition followed by several examples and some examples of functions that are not bilinear.

Reading: In the first part of the lecture we stated Theorem 6 and Corollary 7 in Section 10.3 of Dummit and Foote.  We did not give the full proof of Theorem 6 in lecture, but you should read the details in the book.  We proved Proposition 2.3 in Commutative Algebra by Atiyah and MacDonald.  You can find a scan of the book here. This is an excellent book that covers a lot of the material we have seen in a very direct way.  It also has excellent exercises.  We also mentioned several exercises in Section 10.3 that you should look at.