When I was an undergrad studying abstract algebra, we used the second edition of Artin and covered groups first and then rings. Fields, vector spaces, and algebras came later, I think.

I remember finding group theory difficult because of the number of concepts (cosets, group actions, &c) and the large amounts of only-slightly-different notations for things like orbits and stabilizers. Another thing I remember pretty clearly is being confused by the different meanings of juxtaposition in different contexts (e.g. $gHg^{-1}$ vs $ghg^{-1}$ vs $gX$ where $X$ is a set).

I'm wondering whether there are any books or perhaps courses that don't start with either groups or rings and instead pick something more unusual. I'm not a teacher; my interest in this is entirely theoretical.

Based on this question and this question, it seems like teaching groups or rings first is popular, with rings being firmly in second place.

There are other things though that are perhaps reasonable candidates for an introductory algebraic structure.

  • Fields -- there are a few ready examples, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, and the finite fields. And there's cool stuff like $\mathbb{H}$ if we're willing to consider skew fields.
  • Vector spaces -- row/column vectors with matrices as linear maps and some other examples should be familiar from linear algebra, which is usually taught first.
  • Lattices -- lattices are reasonably straightfoward (in my opinion)

And there are some other things that are weird, but also perhaps reasonable choices.

  • Monoids or semigroups
  • Semirings -- rings without negation.
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    $\begingroup$ In the second paragraph of the question, you list difficulties with group theory. It seems to me that the same difficulties afflict fields and vector spaces as well. $\endgroup$ Jun 23 at 22:48
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    $\begingroup$ @AndreasBlass That's a fair point, but I remember fields and vector spaces being much easier to understand back then and not simply because they were introduced second. I can remove the anecdote if it's extraneous. $\endgroup$ Jun 23 at 23:26
  • $\begingroup$ you could just take a linear algebra textbook that has a suitable "abstract" viewpoint (general base field, vector spaces) for the first part of the course. In Germany, two semesters of linear algebra are typically studied before abstract algebra. $\endgroup$ Sep 21 at 21:15

Not sure if this counts for the OP's purpose, but I started with Hungerford's Abstract Algebra: An Introduction. The first 3 chapters go like this:

  1. Arithmetic in $\mathbb Z$ revisited
  2. Congruence in $\mathbb Z$ and modular arithmetic (i.e., $\mathbb Z_n$)
  3. Rings

Etc. The motivating thesis is explicitly laid out in the Preface and a Thematic Table of Contents -- that it's better to go from well-known concepts to more abstract in fairly small gradual steps (i.e., the reverse of OP's experience with Artin).

Ironically, I now look at a book that starts with groups enviously for its brevity and elegance (rings as a combination of two groups plus distribution), but it seems pretty easy to believe that Hungerford's approach is less of a shock-treatment to students.

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    $\begingroup$ Agreed. I have an old abstract algebra textbook in my home library that spends probably half the book discussing $\mathbb Z$ and then uses that platform to motivate the development of groups and rings. Mind you, that presentation is going to bite any student who goes on to study non-commutative groups, but it was probably fine for someone who would never need to go deeper into the topic. $\endgroup$ Jun 23 at 13:56

A nice example is Paolo Aluffi's Algebra: Chapter 0. The first chapter (40 pages) is Preliminaries: Set theory and categories. Groups and rings follow in the next three chapters.

This book is special as it follows a cathegorically minded viewpoint (in the author's words), which I like very much. According to the introduction, the book is suitable for upper-level undergraduate or beginning graduate courses.


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