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I think I am ready to learn algebra from Lang, but wanted some perspective.

I have been exposed to:

Linear algebra: All of Axler

From my other, legendary honors course:

-Order theory (lattices, partial orders, posets, galois connections)

-Category theory: in the same crazy honors undergraduate analysis course

-Topology: homomorphisms in the category of topological spaces, product topologies, seperation axioms

I have also been exposed to and thought about basic examples of semigroups, involutions, monoids, groups...

My question is, although I own Lang Algebra and absolutely love the style, will it be to my disadvantage to not start with an undergrad book? I suspect that, assuming I'm capable, the graduate treatment would be just fine as long as I'm mature enough. Maybe I'd just get less familiarity with specific examples than I'd get from Artin's book?

Lang himself says in the intro that not having algebra exposure is okay if you have enough mathematical maturity. I'm about to self-study this book, it looks great, I'm just hoping for words of agreement or discouragement (with reasoning!) so that I know it's a good idea.

Lang's book looks so appealing... So please? Can I? =D

NeurallyInspired
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There are many rough spots in Lang's exposition (along with many great insights). But it is easy to remedy these deficiencies. George Bergman has written a superb $\, $ Companion to Lang's Algebra. $ $ Its two-hundred odd pages fill in many of the gaps and provide much supplementary content. Combined you have an algebra textbook written by two leading algebraists - which is quite rare.

That said, I highly recommend that you learn from at least a few different textbooks. You might find it helpful to go to your university library and browse various algebra textbooks to get a feel for the level of exposition, the optional topics treated, the breadth and depth of exercises, etc.

Bill Dubuque
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    Agreed on this. I think if you're prepared for it and like the style, Lang's book is still the best place to learn Algebra, but there are better references for further reading now than the ones he gives. – Callus - Reinstate Monica Jan 02 '14 at 04:10
  • Great, I can't wait. Thanks everyone! – NeurallyInspired Jan 02 '14 at 06:57
  • Dear Bill, I'd like to ask your advice on which AbAlg text I should read next, I would of preferred to contact you through chat, but I'm not sure which room to do that in, so I'll try my best here. I've read Beachy&Blair's Abstract Algebra a couple of times (I enjoyed their group theory exposition) and I've also read the Ring Theory chapters a couple of times in D&F. I spoke with my AbAlg professors, and they suggested Atiyah&Macdonald next or even Bourbaki for Commutative Algebra. – no lemon no melon May 06 '21 at 22:01
  • Perhaps I was too spoiled in Beachy&Blair in how carefully they treat group theory, but I miss the detail that B&B goes into. What would you suggest next? – no lemon no melon May 06 '21 at 22:02
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    @nolemonnomelon I wish I could help but that requires much information that I don't have (e.g. about you, and about books written after my student days a few decades ago). As above, the best I can recommend is to browse the candidates and choose those that match your interests and level of knowledge. Luckily that's much easier to do nowadays than in my student days now that most books are conveniently accessible online. – Bill Dubuque May 06 '21 at 22:30
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    No problem, thanks, I appreciate the response! – no lemon no melon May 06 '21 at 23:28