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.

I recommend this Modules and Rings by Dauns, I also mention that is for modules in general, the ring isn't necesarily commutative, but covers what you mention.

I'll preface this answer with the remark that the text I'm recommending probably isn't for most people, but if it suits your style it will provide a clear understanding of the material. Now for the specifics:

I found the first chapter of "A Course in Homological Algebra" by Hilton and Stammbach to be a good introduction. It's not as detailed or long as most texts (principally because it's a single chapter in a book whose focus is not modules) but (at least for me) it provides an excellent framework for the subject, and from there texts like Dummit & Foote or Atiya-MacDonald may be more accessible. Since it's geared towards homological algebra it introduces sequences of modules and the categorical viewpoints of certain constructions (e.g. (co)products) earlier than many other texts. In general it does a good job motivating the concepts.

What I think sets this book apart from others are the exercises. Many are veiled statements of common theorems (most memorably the splitting lemma, Schanuel's lemma, and a classification of hereditary rings that immediately leads to the Fundamental Theorem of Finitely-Generated Modules over a PID) that not only require you to think but also have significant uses outside the book (i.e. they're not obscure results that you'll never use again).

[I apologize if this post is too far afield, but I feel like I should least put this out there on the off chance that this will help]

Martin Isaac's Algebra: A Graduate Course is pretty good for a basic introduction to modules, although he has a kind of unusual group-operator-theoretic approach to it. Looking back I found it very helpful (although at the time I remember thinking I disliked his exercises a lot.)