Comments on Projective Modules

In this handout we will briefly explore the topic of projective modules in a bit more detail than we covered in class. Throughout R is a commutative ring. Recall that, by definition, a projective module is an R-module that is a direct summand of a free R-module. As mentioned in class, if the ring R is decomposable, e.g., R = R1 ⊕R2 is a direct sum of rings, then there are many examples of non-free projective R-modules in particular, R1 is a projective Rmodule that is not a free R-module. When R is indecomposable, in particular, when R is an integral domain, it is much more difficult to gives examples of projective modules that are not free modules. (Of course, over certain classes of rings, projective modules are always free modules). The purpose then of this hand out is two-fold. We first present two classical examples of integral domains admitting non-free projective modules. Our second purpose is to elaborate on the locally free property of projective modules. We saw in class that if R is a Noetherian domain, then a finitely generated locally principal ideal is a projective R-module. In fact, this holds in much greater generality as we will see below.