Methods of graded rings / Constantin Nastasescu, Freddy Van Oystaeyen

The book is a clearly written exposition of the theory of graded rings; in general the categorical point of view is used. Several important functors, the forgetful functor, its right adjoint and the induction and coinduction functors, which stem from representation theory, are shown to be very useful.
The first three chapters of the book are dedicated to the study of the general properties of the category of graded rings and the category of graded modules (in particular the structure of simple objects and injective objects in this category are considered) and the study of strongly graded rings, a fundamental class of graded rings.
In Chapter 4, the methods in the theory of graded rings begin with the consideration of the graded Clifford theory. This method is inspired by the classical Clifford theory of group representation theory. The idea is to understand the structure of a gr-simple module when viewed as an ungraded module. Several applications of this method are presented, a density theorem for gr-simple graded modules is obtained and the Jacobson radical for G-graded rings, in particular when G is either a finite group or an abelian torsion group, is studied.
In the next chapter, the technique of internal homogenization for rings graded by ordered groups is studied. This technique is very useful to investigate chain conditions and Krull dimension of graded rings and modules.
Chapter 6 introduces external homogenization, a technique stemming from algebraic geometry, and Chapter 7 deals with smash products. This construction is inspired in the concept of smash product associated to a Hopf module algebra. Chapter 8 presents the localization theory for graded rings and modules that includes a recent graded version of Goldie's theorem.
In the last chapter some applications of the machinery developed to the problem of gradability, whether we can introduce a G grading on a module making it a graded module, are made.
Moreover, many good exercises enhance the text. This book will be very useful for graduate students interested in the topic. For researchers it makes a quick and accessible reference to the material covered. (MathSciNet)