Texts for direct limit of algebraic structures

Question

I've been studying direct limits with Aityah and Macdonald's textbook, but they're in the form of exercises and I don't feel very confident with them. Does anyone have texts that discuss the direct limits of algebraic objects (modules, rings, algebras) without going into category theory?

Thanks!

Answer 1

Rotman's 'Introduction to Homological algebra' second edition has quite a lot on them, with a particular emphasis on modules, although there is some basic category theory in there.

For commutative rings Appendix A of Matsumura's 'Commutative Ring Theory' covers enough on direct and inverse limits.

Enochs and Jenda's 'Relative homological algebra' also has plenty on both direct and inverse limits, and that's exclusively with modules.

There are going to be plenty more, and a lot of online resources as well.

Answer 2

I don't know of any specific references, but I would recommend looking at examples. One helpful clue is that the tensor product is an example of a direct limit. Given two rings and no specific morphisms between them, their direct limit is their tensor product over $\mathbb{Z}$. Given three rings and two morphisms $A\leftarrow C\rightarrow B$, the direct limit is $A\otimes_CB$. For example, $R[x]\hookleftarrow R\hookrightarrow R[y]$ gives the direct limit $R[x]\otimes_RR[y]\simeq R[x,y]$.