It turns out that when it comes to infinite groups/modules, some algebraic concepts are deeply connected to the underlying set theory (for example, the notion of freeness, the structure of Ext, etc). A good reference for this subject is the book "Almost free modules" by Eklof and Mekler. This book introduces the works of Shelah, Gobel, Eklof and many other important contributors in this field. This research has also led to some interesting developments in "pure" set theory, such as the introduction of black-boxes by Shelah (some diamond-like combinatorial principles which can be proved in ZFC alone, and allow the construction of many interesting algebraic objects).
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