27 votes

Rings before groups in abstract algebra?

My favorite textbook for an undergraduate course in Abstract Algebra, Ted Shifrin's Abstract Algebra: A Geometric Approach, uses a rings-first approach. The primary pro is that students are much more ...
Michael Joyce's user avatar
18 votes

Is MacLane and Birkoff's "Algebra" suitable today as either an undergraduate or graduate text in abstract algebra?

Preface. Birkhoff & Mac Lane's Algebra is a brilliant book. I should probably spend some time with it again, actually. Also, I apologize for such a long response. I think too much about algebra ...
Joseph R. Heavner's user avatar
18 votes

Rings before groups in abstract algebra?

I have taught both groups first and a rings first course. When I was a post-doc at Rutgers University, I taught their standard introduction to modern algebra course using Hungerford's undergraduate ...
Bill Cook's user avatar
  • 646
16 votes

Simple examples that violate group axioms

Combining colored paint is an interesting example of a non-associative operation. Define $Paint_1 * Paint_2$ to be the paint obtained by mixing the two paints in a $1:1$ ratio. It is easy to see that ...
John Coleman's user avatar
  • 1,536
15 votes

Examples of basic non-commutative rings

The quaternion ring is a pretty simple example of a non-commutative ring (a skew-field, even).
David Steinberg's user avatar
15 votes

Rings before groups in abstract algebra?

I've done it both ways, although I do rings-first now and for the foreseeable future. I think the pros and cons have a lot to do with the audience, especially if there are a lot of pre-service ...
Robert Talbert's user avatar
14 votes

What are the "best" groups to use as examples while learning new concepts in algebra?

Symmetry groups of basic regular polygons and polyhedra. For one thing, they are noncommutative; for another, sometimes they coincide with other "known" groups; for a third, there can be physical ...
kcrisman's user avatar
  • 5,980
14 votes

The term "unique" for functions and operations

I don't think there's a lot of educational value to fixating on trying to word definitions in exactly the perfect way. Students have trouble with the notion of a function because it's hard. The way ...
Henry Towsner's user avatar
11 votes

Simple examples that violate group axioms

I really like the example of the game Rock-Paper-Scissors, thinking of it as a binary operation $\star$ on the set {Rock, Paper, Scissors}. The rules are: Rock$\star$Paper $\mapsto$ Paper Rock$\...
Nick C's user avatar
  • 9,436
11 votes
Accepted

How to teach abstract algebra for the first time?

I second Adam. Keep it with strong with examples if you can. There is a mathoverflow answer with a ton of group theory examples. You might want to force them to practice doing more "rote" exercises ...
Chris C's user avatar
  • 2,594
10 votes

Examples of basic non-commutative rings

Why not turn $M_2(\mathbb{R})$ into a multiplication on $\mathbb{R}^4$? Here's a fun, somewhat weird, ring: $$ \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \left[ \begin{array}{...
James S. Cook's user avatar
10 votes

Where do students learn to solve polynomial equations these days?

Students learn linear and quadratic equations in high school algebra. And then, if they have forgotten it, re-learn it in college, in courses called "pre-calculus" or something. Unless they ...
Gerald Edgar's user avatar
  • 7,489
10 votes
Accepted

Requesting a Polynomial System of Equations

Here's a really nice textbook chapter that covers using algebraic geometry to compute equilibria in economics. It looks pretty approachable for students while simultaneously getting into plenty of ...
Justin Skycak's user avatar
9 votes

At what point in the curriculum should the tensor product be introduced?

(This pertains to U.S. universities.) Times may have changed, but when I was in graduate school (several places, and yes I know this is unusual, but I mention it because I'm talking about more than ...
Dave L Renfro's user avatar
9 votes
Accepted

Where do students learn to solve polynomial equations these days?

The rational root theorem, synthetic division, the remainder theorem, Descartes rule of signs, and similar lower level topics were fairly widely taught in U.S. high school algebra-2 courses before and ...
Dave L Renfro's user avatar
9 votes

Introductory real analysis before or after introductory abstract algebra?

Despite the names of these fields, as a student I found real analysis more abstract than abstract algebra: real analysis was less real and more abstract to me than abstract algebra. I don't think I ...
Joseph O'Rourke's user avatar
8 votes

What are the differences between popular undergraduate abstract algebra books?

Judson's Abstract Algebra: Theory and Applications is different in that it is an open source textbook that is available at no cost. I haven't used it (yet), but I think it's worth pointing out for the ...
J W's user avatar
  • 4,653
8 votes

Examples of basic non-commutative rings

How about $\mathbb{Z}[i]$ together with conjugation, thought of as a ring element? Conjugation and multiplication by $i$ do not commute.
Adam's user avatar
  • 5,683
8 votes

How to teach abstract algebra for the first time?

I think that you probably need to give them some hands-on experience before any of it will sink in. I assume that they are CS students? After you give them the definition and a few examples, you could ...
Adam's user avatar
  • 5,683
8 votes

Introductory real analysis before or after introductory abstract algebra?

Is 'at the same time' an option? I mean, by junior year, math majors should be taking at least two math classes per semester, right? When I was an undergraduate at Penn State, these two courses were ...
Joe's user avatar
  • 333
8 votes
Accepted

Undergraduate-level abstract algebra books or courses that don't start with groups or rings

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: Arithmetic in $\mathbb Z$ revisited Congruence in $...
Daniel R. Collins's user avatar
7 votes

Rings before groups in abstract algebra?

I think groups first is the right approach. My reasoning is that, groups are very "strange" structures in that they don't feel natural to work with at first. Rings are much more intuitive because as ...
Alex Mathers's user avatar
7 votes

Simple examples that violate group axioms

For something that's not commutative, try rotations about axes through the origin in 3D. A simple rectangular book can illustrate this very well. Try rotating the book in 90 degrees around vertical ...
DavidButlerUofA's user avatar
7 votes

What are the "best" groups to use as examples while learning new concepts in algebra?

You mentioned the linear groups over the real numbers, which I think is a useful example, with the potential drawback that it is infinite. To "remedy" this one can consider the linear groups over ...
quid's user avatar
  • 7,692
7 votes

Simple examples that violate group axioms

I can't believe no one has mentioned (positive, say) integers under exponentiation as a non-associative operation. It's by far my favorite, though of course this example is also not commutative and ...
kcrisman's user avatar
  • 5,980
7 votes

Where do students learn to solve polynomial equations these days?

In the United States, solving linear and quadratic equations is a standard part of Algebra 1, which most students take in 8th or 9th grade. Students will return to polynomials and see long division ...
Tim Ricchuiti's user avatar
7 votes

in what sense is the subject of finite group theory 'algebraic'?

Typically, students learn about finite groups long before they think of "variety of algebras" or "compactness theorem". I suppose you also want to exclude fields from "...
Gerald Edgar's user avatar
  • 7,489
7 votes
Accepted

Third isomorphism theorem: how important is it to state the relationship between subgroups?

I agree that $\text“\color{red}{N \text{ is a normal subgroup of a group }H}\text”$ is a redundant/bogus condition of the given theorem. To specify what is rightly a $\color{red}{\textrm{lemma}}$ of ...
ryang's user avatar
  • 1,822
7 votes

Introduction of group action as morphism of groups

I prefer just to keep the language as simple as possible and the level of abstraction as low as the circumstances allow when teaching courses for students who see the material for the first time. So ...
fedja's user avatar
  • 3,809
6 votes

What are the "best" groups to use as examples while learning new concepts in algebra?

In addition to the dihedral groups, the quaternion group $Q_8$, the symmetric and alternating groups, and small matrix groups, I have always found the following examples useful: Automorphism groups ...
Jim Belk's user avatar
  • 8,269

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