I’m signed up for a first-year grad algebra class and the assigned textbook is Dummit & Foote. So I’m looking for another book or two that would go good with D&F.
Some qualities I’m looking for: (0) Begins with groups or categories (or semigroups or monoids), (1) Strong treatment of groups and representations of finite groups, (2) Strong treatment of noncommutative rings, (3) Doesn’t shy away from category theory, and (4) Covers some homological algebra.
Bonus points if it talks about Lie algebras and k-theory.
I’m not opposed to older or terse prose.
Oh, and I’m not a fan of Hungerford’s “Algebra.”
Edit: I am not looking for a category theory book. Nor am I looking for a book which covers everything completely through category theory. But I’d like it covered.
I would consider Algebra: Chapter 0 by Aluffi, although it doesn’t satisfy all your requirements. The voice of the text is light and conversational, and Aluffi does give some allusions to analogies in other interesting categories at times (e.g. category of smooth manifolds).
The book’s language is highly categorical — Chapter I is entirely dedicated to introducing categories, and the categorical theory develops as the algebraic does.
Groups are well-represented, with Chapter II and IV fully dedicated to group theory, although (as Aluffi points out in the introduction) any representation theory is “missing altogether.”
While I wouldn’t say that noncommutative rings are strongly treated, they are certainly treated well. Chapter IX focuses fully on homological algebra.