Algebra II Ring Theory Vol. 2: Ring Theory

From Math This is the second in a two volume set. It is composed of ten chapters labeled 17 through 26 (the first 16 chapters are in Volume 1). Chapter 17 studies the lattice of submodules of a module over an associative ring, with such theorems as the Jordan-Hölder, Fittings, and Levitzkis'. Chapter 18 begins by introducing the Jacobson radical of a module, small submodules, decomposition into direct sums à la Remak, Krull, Schmidt and finishes with semi-perfect rings. The material in Chapter 19 concerns quasi-injective modules and their endomorphism rings. It is in this setting that the Jacobson density theorem is given, von Neuman regular rings, rational extensions and maximal quotient rings are introduced, and the Goldie theorems are proved. QI rings are also studied in this chapter. Chapter 20 begins with the theorems on how the chain conditions relate to the cardinality of the summands of direct sum decompositions of injectives. Next Chatter's theorem on the decomposition of Noetherian rings into semiprime and Artinian rings is given. The chapter concludes with a section on valuation and almost maximal valuation rings. Chapter 21 deals with direct sums of modules, each of which has a local endomorphism ring, hence Azumaya type uniqueness theorems on summands of infinite direct sums of indecomposable modules. Chapter 22 is devoted to perfect rings. The subject of dualities between module categories is extensively studied in Chapter 23. The basic results on quasi-Frobenius rings are given in Chapter 24. Chapter 25 begins with Warfield's structure theorem of serial rings, then gives Nakayama's structure theorem for Artinian serial rings (generalized uniserial). Next is a look at rings for which every finitely generated module is the direct sum of cyclic modules and finally the Eisenbud-Griffith-Robson theorem. Chapter 26 contains the classical descriptions of the prime and Jacobson radicals. It also contains Amitsur's theorem.