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I'm now working on the exercises of Chapter 2 of Atiyah-Macdonald, and I consult some algebra books, for example, Lang's Algebra and Aluffi's Algebra: Chapter 0, to help me understand the things like tensor products and flatness.

However, these book all use categorical concepts like direct limit, adjoint pair heavily to prove the basic propositions of flatness, for example, the preserving of flatness in direct sums.

So I open some category textbook and try to learn some stuffs like adjoint pair. But they are too abstract for me that I spent whole days only learn category, without any progress on commutative algebra.

So my question is: Is there any textbook teaches the proofs of basic properties of tensor products and flatness in detail and doesn't use the categorical concepts like adjoint pair?

I know that Atiyah-Macdonald itself didn't use any category stuffs, but it didn't give much details in proofs and a lot of key propositions were in exercises. I'm learning these concepts for the first time so it's difficult for me to go through.

Thanks!

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    $\begingroup$ You can take a look at the first chapter of Bourbaki's Commutative Algebra. $\endgroup$ Jun 27, 2017 at 10:12

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See abstract algebra by Dummit Foote . The concepts are explained in detail. In particular, it's treatment of tensor product is based on arbitrary rings and not on commutative rings like in Atiyah. It has also has an appendix on Category and functors which is explained in an easy style.

By the way flat modules are explained just after the chapter on tensor product along with projective and injective modules which Atiyah Don't have.

Exercise of Tensor product is highly recommended from that book.

Also I should mention that in the first two pages of the chapter tensor product it motivates the need to introduce the concept of tensor which Is great and which most of the books doesn't write in detail.

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