Books for linear algebra over commutative rings
Solution 1:
J. S. Milne has a lengthy expository note on commutative algebra on his homepage: http://www.jmilne.org/math/xnotes/ca.html.
As the abstract says, this is written on quite a high level, though:
These notes prove the basic theorems in commutative algebra required for algebraic geometry and algebraic groups. They assume only a knowledge of the algebra usually taught in advanced undergraduate or first-year graduate courses. However, they are quite concise.
Solution 2:
A concise and excellent treatment of what you want can be found in Lang's "Algebra", in the chapter "Matrices and Linear Maps". The tensor product is also given in terms of modules.
Solution 3:
You also have the specific Linear algebra over commutative rings by B.R. McDonald (1984). Its contents are:
- Matrix theory over commutative rings
- Free modules
- The endomorphism ring of a projective module
- Projective modules
- Theory of a single endomorphism