Introduction

At this point the reader might wish to refamiliarize her or himself with the basic Wedderburn theory, covered in 7–1.35 through 7–1.43. Since we first use projective modules in this chapter (in Theorem 10–4.4), they are discussed at the beginning of this chapter. The reader who is mainly interested in the refinements and additions to Wedderburn theory of this chapter, need only read 10–1.1 through 10–1.13. Since projective modules are important in their own right, and since we need them in many subsequent chapters, the rest of the elementary theory of projectives is also developed in the first section of this chapter.

Projective modules are generalizations of free modules, because a free module is projective. For this reason logically, in successively studying more and more general classes of modules, they could very well be studied right after the free modules and before the injectives. In our treatment of free, projective, and injective modules we are assuming the ring R has an identity 1 ≠ 0 and all modules are unital. However, from then on in sections 10–2 throughout 10–7, we most decidedly do not assume that the ring has an identity. This chapter can be looked upon as being an improvement and extension of previous more basic material – first, the projective modules generalize the free ones, and then we develop in more detail the fundamental Wedderburn theory already covered in 7–1.35 through 7–1.43.