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I am interested in topology, while I am not so comfortable with some algebraic flavored textbook on algebra. Actually, it was not until I learned some topology that I began to understand some abstract algebra. I believe that behind every algebraic theorem, there is a geometric analogue, and this is what I am interested in.

I want to find algebraic textbooks that are of geometric flavor. Say, Armstrong's book, Groups and Symmetry is a lovely book on group theory that is of this type. Are there some more such textbooks (on ring theory and homological algebra)?

Thank you!

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    $\begingroup$ Does algebraic geometry count for the geometric flavor you're looking for? $\endgroup$
    – Fernando Muro
    Commented Jan 5, 2015 at 9:23

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I guess it depends what you consider geometry. I think Eisenbud's book Commutative Algebra with a View Towards Algebraic Geometry is magnificent, and motivates commutative algebra from stem to stern by means of the connections with geometry. I wish it had existed when I was a graduate student. But it's algebraic geometry, not algebraic topology, for sure.

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    $\begingroup$ I think this should be taken with a grain of salt. A lot of the motivation in Eisenbud may only be "magnificent" after you have some understanding of the algebraic geometry involved. $\endgroup$
    – Charles Staats
    Commented Jul 27, 2010 at 15:13
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Confusingly, in addition to Emil Artin's Geometric Algebra mentioned by Anweshi, there is Michael Artin's textbook Algebra. The latter is more of a modern first-year graduate textbook covering the usual topics you'd find in a book like Lang, Dummit and Foote, and so on. I don't have Michael Artin's book handy but I remember a lot of it having a "geometric flavor," e.g. there was a lot of discussion of symmetry groups and linear representations.

This was one of the first abstract algebra books I read and I remember loving it at the time.

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  • $\begingroup$ Couldn't have agreed more! $\endgroup$
    – Somnath Basu
    Commented Jul 27, 2010 at 3:55
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    $\begingroup$ Michael Artin's book is a nice read, but I wouldn't quite put it at the first-year graduate level. $\endgroup$
    – Elizabeth Henning
    Commented Jan 5, 2015 at 7:55
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Try Artin's Geometric Algebra.

You might also find interesting Shafarevich's book “Basic Notions of Algebra”, for understanding the philosophy behind algebra. I strongly recommend that you spend some time with that book.

Also try Coxeter's various books such as Projective Geometry, etc..

But of course if you really want to understand the "geometry behind the algebra", then you should first have a good footing in algebra, and then you should study algebraic geometry. Even for Artin's Geometric Algebra, you better first have an understanding of algebra before reading it. It appears to me that at the moment the best bet for you will be Shafarevich's "Basic Notions of Algebra".

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    $\begingroup$ Oh my god, "Basic Notions of Algebra" is fantastic! I always thought that this type of book does not exist yet and I already started to write something like that :-). $\endgroup$
    – Martin Brandenburg
    Commented Jul 27, 2010 at 7:13
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    $\begingroup$ @Anweshi: Shafarevich is the only author of the book “Basic Notions of Algebra”. $\endgroup$
    – Dmitri Pavlov
    Commented Jul 27, 2010 at 11:08
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    $\begingroup$ @Dmitiri Pavlov: Such was my feeling too. But before making the answer I looked in Amazon and saw this: amazon.com/Notions-Algebra-Encyclopaedia-Mathematical-Sciences/… $\endgroup$
    – Anweshi
    Commented Jul 27, 2010 at 12:59
  • $\begingroup$ @Martin: :-) ... $\endgroup$
    – Anweshi
    Commented Jul 27, 2010 at 13:02
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    $\begingroup$ Dmitri is right; Kostrikin is one of the editors of the series the book appeared in. Speaking of Kostrikin, he coauthored "Linear Algebra and Geometry" with Manin. $\endgroup$
    – Franz Lemmermeyer
    Commented Jul 27, 2010 at 13:21
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You may also enjoy "Visual Group Theory" by Nathan Carter.

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    $\begingroup$ There's also the associated software, Group Explorer, at groupexplorer.sourceforge.net. It can be used independently of the book. $\endgroup$
    – J W
    Commented Jul 27, 2010 at 10:36
  • $\begingroup$ This is my favorite book on this subject too. If only I had had it when I took modern algebra in college! $\endgroup$
    – David G. Stork
    Commented Jan 14, 2019 at 7:30
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One fundamental aspect of the connection between geometry/topology and algebra is the analogy between galois theory of fields and covering theory in algebraic topology and algebraic geometry. A nice book on the subject has appeared recently : Tamas Szamuely's Galois Groups and Fundamental Groups.

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    $\begingroup$ Also Askold Khovanskii, Galois theory, coverings, and Riemann surfaces. $\endgroup$
    – Ben McKay
    Commented Jan 14, 2019 at 14:30
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manin's linear algebra and geometry

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