Introduction to Volume I
A foreword supplies the necessary background in the theory of sets and cardinal (ordinal) numbers, and the concepts of group (and the Noether homomorphism theorems), semigroup, category, functor are introduced in Chapter 1. Products, coproducts, and the free object functors in various categories constitute Chapter 2. Rings, modules, ideals, and the Baer-Eckmann-Schopf notions of injectivity, essential extension, and injective hull are introduced in Chapter 3. Chapter 4 contains an elementary presentation (without tensor products) of the Morita theorems characterizing category equivalence of categories of modules over two rings. The basis for this is the correspondence theorem for projective modules, from which also is derived most of the known structure theory of noncommutative Noetherian simple rings. Chapters 5 and 6 survey limits, limit preserving functors, additive and abelian categories, free algebras, (semi) group and polynomial algebras, and the adjoint functor theorems of Eilenberg, Freyd, Gabriel, Kan, Mitchell, and Watts. These six chapters comprise Part I.