This is an elementary introduction to some aspects of ring theory and homological algebra. The reader is just assumed to have had a basic course in algebra including some acquaintance with rings.
Modules and morphisms between them are introduced, rather tersely, in Chapter one, including the basic operations: Intersections, quotients, direct products and sums, kernels, images, and exact sequences. Three classes of modules are also defined: simple, projective and injective, with some elementary properties of these classes quickly proved, from Schur’s lemma to characterization of projective or injective modules in terms of splitting exact sequences. Since all of this is done in just 10 pages, the reader with just a basic knowledge of groups and rings may find this chapter overly terse.
Chapter two is devoted to the proof of the Artin-Wedderburn theorem: every semisimple artinian ring is isomorphic to a direct product of finitely many matrix rings over division rings. Several homological statements are shown to be equivalent to the ring-theoretical Artin-Wedderburn decomposition theorem, paving the way for the introduction to homological methods in Chapter three: complexes, homology, projective resolutions and the Ext-functor, culminating with the characterization of projective modules in terms of the vanishing of the corresponding Ext-modules.
This last result is used to motivate the various homological dimensions introduced in Chapter four. It is in this chapter where we find the characterization of injective modules in terms of the Ext-functor and the existence of enough injectives in the category of modules. Chapter four includes several applications to ring theory, for example to show that for a left and right noetherian ring, the left and right global dimensions agree. Chapter five applies all this machinery to duality theory and quasi-Frobenius rings, culminating with a homological characterization of these rings.
This is a charming book, with limited but attainable goals, within the reach of an advanced undergraduate or beginning graduate student. There are, of course, more complete texts on homological algebra, but this little Dover reprint is a reminder that the subject is always close to some of its roots: rings and algebras.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.