First published in 1991, this book contains the core material for an undergraduate first course in ring theory. Using the underlying theme of projective and injective modules, the author touches upon various aspects of commutative and noncommutative ring theory. In particular, a number of major results are highlighted and proved. Part I, "Projective Modules", begins with basic module theory and then proceeds to surveying various special classes of rings (Wedderburn, Artinian and Noetherian rings, hereditary rings, Dedekind domains, etc.). This part concludes with an introduction and discussion of the concepts of the projective dimension. Part II, "Polynomial Rings", studies these rings in a mildly noncommutative setting. Some of the results proved include the Hilbert Syzygy Theorem (in the commutative case) and the Hilbert Nullstellensatz (for almost commutative rings). Part III, "Injective Modules", includes, in particular, various notions of the ring of quotients, the Goldie Theorems, and the characterization of the injective modules over Noetherian rings. The book contains numerous exercises and a list of suggested additional reading. It is suitable for graduate students and researchers interested in ring theory. Readership Graduate students and research mathematicians interested in ring theory. Reviews ""There seems to be an emerging consensus as to what material should constitute the core of a first course in moduletheoretic ring theory ... The book ... is definitely within the bounds of that consensus ... presentation is clear, the proofs are often quite ingenious and the exercises are well chosen ... definitely suitable for use as a textbook.""  Mathematical Reviews "This excellently written book, which has been published originally by Wadsworthand Brooks in 1991, is already a classic ... A book recommendable now as before!"  Monatshefte für Mathematik Table of Contents Projective modules  Modules and homomorphisms
 Projective modules
 Completely reducible modules
 Wedderburn rings
 Artinian rings
 Hereditary rings
 Dedekind domains
 Projective dimension
 Tensor products
 Local rings
Polynomial rings  Skew polynomial rings
 Grothendieck groups
 Graded rings and modules
 Induced modules
 Syzygy theorem
 Patching theorem
 Serre conjecture
 Big projectives
 Generic flatness
 Nullstellensatz
Injective modules  Injective modules
 Injective dimension
 Essential extensions
 Maximal ring of quotients
 Classical ring of quotients
 Goldie rings
 Uniform dimension
 Uniform injective modules
 Reduced rank
 Index
