Some basic book to start with modules? I am very interested in studying modules, which I have studied algebra, is basic theory of groups and rings, of Hungerford and Dummit. I was reading the Dummit the module, but I still struggled a bit in the section and the tensor product of exact sequences, as in the book of Atiyah. They know a simple book to begin with?
The best book I've ever seen on the topic is Module Theory: An Approach To Linear Algebra by T.S. Blyth. The book is extremely rigorous yet gentle and can be read after an undergraduate abstract algebra course. That's the book I'd start with. 

Another book you might want to take a look at is Algebra: An Approach Via Modules by Steven Weintraub and W. Adkins. This is a very unusual and nearly forgotten graduate algebra text that presents a standard first year graduate course in algebra with total emphasis on module theory and how linear algebra and its generalizations unify algebra. I found the book very nicely written with a ton of concrete examples-it also has an outstanding final chapter on group representations. But to me, it's too classical and easy for a graduate course.There's no discussion of category theory and very little homological algebra. And the authors deliberately leave out field theory for another book (Weintraub's Galois Theory). I DO think it's a nice book to have and I'd consider using for an honors algebra course instead of Herstein or Artin. For your purposes, I think it's worth a very careful look since the book's overriding theme is module theory and how it relates all of algebra.  

Those are the 2 books I'd use.
Sorry at first I didn't notice the rest of your text. you can read (Robert B. Ash, Abstract Algebra) and (herstein).