Good book on linear algebra over rings (i.e. modules)

[eof]

Can anyone recommend a book that covers linear algebra through the perspective of modules? I am basically trying to find something that would highlight all the differences between modules and vector spaces.

Lam has written the book Lectures on Rings and Modules, which is good, but doesn't really fit this purpose. It's more geared towards the module theory you need for homological algebra (i.e. injectives, projectives etc.).

 

[jbunniii]

Have a look at Roman's "Advanced Linear Algebra" if you haven't already. You can see the table of contents and look at excerpts at this Google Books link if you scroll down a bit:

http://books.google.com/books?id=bS...resnum=3&ved=0CBsQ6AEwAg#v=onepage&q=&f=false

 

[Landau]

Many Abstract Algebra books talk about modules and apply it to vector spaces. E.g. Dummit and Foote.

 

[mathwonk]

A brief treatment is given in my free course notes for math 8000[6] on my web page:

http://www.math.uga.edu/~roy/

these were actual notes for a graduate course in algebra lasting one semester and intended to prepare students fior the PhD prelim in algebra. (It succeeded for about half of them.)

Another treatment that does not mention modules, intended for advanced undergraduates is given in my notes on that same page, for math 4050.

A more detailed treatment using modules, is given in my notes on that same page for math 845. the ring theory is given in the math 844 notes. these (843-4-5) were also actual class notes for a graduate course back when the course lasted 3 quarters. thus they contain more detail and are perhaps more useful.

Actually I have four treatments of linear algebra on that page, at almost any length you wish:

from longest to shortest, the first two using modules:
math 845,
math 8000[6],
math 4050,
primer of linear algebra (15 pages!)In published form, a standard reference is Lang, Algebra, the section on decomposition of modules over a pid.