Abstract: It is well-known that finitely generated projective modules over a local ring are actually free modules. However, projective modules over non-local rings need not be free, i.e., the property that a module has a basis over a ring is a property that cannot be checked locally. Serre famously proved that a projective module with rank larger than the Krull dimension of the ring it is projective over necessarily contains a free direct summand. Serre's result can be reinterpreted as saying indecomposable projective modules necessarily have small rank. We will discuss the significance of Serre's result and provide an elementary proof in the case that our ring is affine over an algebraically closed field. We will also see how Serre's Splitting Theorem can be generalized to all finitely generated modules and provide an elementary proof of the Generalized Serre's Splitting Theorem, a result originally proven by Stafford.