I'm looking for a book that provides a nice introduction to Modules for a student that already had a first course in Abstract Algebra in groups and rings. The book should explain why are modules useful and what can they be used for.
Thank you in advance!
I'm not aware of any text that deals solely as an introduction to modules, since they're often introduced in more general texts on ring theory or algebra. This isn't really aided by the fact that they occur in the most arbitrary and unexpected places.
If I were to give an advise on this matter, I wouldn't recommend looking for a specific introductory text, really. Authors tend to write in such a way that their own points of view seem the natural ones and that their own approach to the subject seems the natural one. If the subject is a broad and versatile one, like is the case with modules, this might restrict one at a later point. I'd therefore advise to first read the definition and some of the basic properties of modules in some book on abstract algebra - any book called something like 'introduction to abstract algebra' should contain at least one section on modules, containing the material I'm referring to. After that, I'd recommend reading about several different places in mathematics where modules occur, with the emphasis on relating this back to the basic concept, and seeing why and how they occur there. One aspect that should be included in this, is representation theory and I'd say lattices (possibly crystallographical applications) shouldn't be excluded either. Other users will most likely be able to mention some other subjects as well.