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I was thinking about reviewing linear algebra to recover many theorems that I can use over commutative rings with unity. But it seems very tedious and I did not want to make any mistakes on these theorems, as I often need to use them. I am wondering if there are good books out there for this purpose and want to know why they are good.

More specifically, I want a good book that discusses (finite size) matrices over ring and their relationships with $R$-module homomorphisms, where $R$ is a ring or commutative ring (with $1$, of course).

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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.

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I have a copy of Lang, so +1! I will check it out when I go home. – user123454321 Feb 9 '13 at 20:30

J. S. Milne has a lengthy expository note on commutative algebra on his homepage:

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.

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Thanks for the wonderful link, but I don't think this is what I asked, although I needed to ask this question to study CA. – user123454321 Feb 9 '13 at 20:32

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