Book for Module Theory

I want a book to cover the following topics in Module Theory:

Modules, Submodules, Quotient modules, Morphisms Exact sequences, three lemma, four lemma, five lemma, Product and Co products, Free modules, Projective modules, Injective modules, Direct sum of Projective modules, Direct product of Injective modules.

Divisible groups, Embedding of a module in an injective module, tensor product of modules, Noetherian and Artinian Modules, Finitely generated modules, Jordan Holder Theorem, Indecomposable modules, Krull Schmidt theorem, Semi simple modules, homomorphic images of semi simple modules.

I want a book that covers these topics. It is not that a single book should contain all of these but it would be better if it is so. The book should be interesting, have some good problems to work on (would be better if it is provided with hints to hard problems).

Comments and suggestions are needed.

Solution 1:

As it was suggested before, Module Theory: An Approach to Linear Algebra by T. S. Blyth is an awesome title which covers almost every basic topic of Module theory in a very elegant, clear and efficient way. It is hands down my favorite text in the subject, but unfortunately it has been long out of print and therefore it is expensive and hard to obtain.

I contacted T. S. Blyth a few months ago looking at the possibility of a new edition, or at least, for a Dover reprint or something like that. Recently, Tom told me that he just finished an electronic edition, and I inmediately helped him with a careful and exhausting typo hunting. With great joy, I shall let you all know that this venerable text is now available for free!

Download the book here!

Solution 2:

I recommend "Basic Algebra" (I and II) by Nathan Jacobson.

Volume I covers all the basic notions (Modules, Submodules, Quotient modules, Morphisms, finitely generated modules). Volume II covers Noetherian and Artinian Modules, Jordan Holder Theorem, Krull Schmidt theorem, injective and projective modules, tensor product of modules, etc.

Volume II contains also chapters on category theory and universal algebra (relevant for notions like Product and Coproduct that you mentioned).

These books are really nice and clear, and contain a lot of other basic material in almost every subdomain of algebra I can think of.