Rings before groups in abstract algebra?

The default approach to teaching abstract algebra seems to be groups first, then rings. However, occasionally a textbook pops up (e.g. Childs' A Concrete Introduction to Higher Algebra, Hodge et al's Abstract Algebra: An Inquiry Based Approach or Aluffi's recent Algebra: Notes from the Underground) that starts with rings, based on the familiar examples of integers and polynomials.

What are the pros and cons of taking a rings-first approach?

• +1 I was wondering about this recently, also, so am very interested in what answers are given. First, polynomial rings seem like a familiar palatable object to students (could be a `pro') but the appropriate way to view these objects abstractly (I'm thinking e.g. of things like evaluation homomorphism) may be tough to sell because students feel they "already understand" polynomials. Quotients in these rings also seem simpler at first glance (just think the polynomial generating the ideal is always zero in the quotient!) I'm not sure the students really get this though. Jan 29, 2016 at 15:25
• Side comment: in any case, I would advise to only introduce these structures when one has sufficiently many examples at hand to motivate the abstract notion. Integers and polynomials are enough to introduce rings, but often groups are introduced before one has good examples to show ; studying permutations before abstract groups thus seems a good idea. Jan 29, 2016 at 21:57

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 familiar with examples of rings (integers, polynomials) than they are with the standard examples of groups (symmetries of simple shapes, permutations). Indeed, the groups that students are familiar with tend to be the additive groups of known rings and the multiplicative groups of non-zero elements of known fields. Another pro is that the extra structure in a ring allows students to have more tools to work with when constructing proofs.

The primary con of this approach is that groups have a simpler list of defining axioms, so proofs are easier in the sense that there are less wrong directions to veer off into. Another con of the rings-first approach is that it limits how much time you can spend on groups in the first-semester course. To get to interesting group theory (in-depth studies of symmetry / Sylow Theorems / classification of finite groups of small order), one generally needs an entire semester. Doing groups first allows an instructor to do that in the first semester, and then the second semester can be devoted to rings and fields and an introduction to Galois theory (or, of course, one can go in other directions after introducing rings).

I should add that in Shifrin's book, rings-first really means commutative rings with unity first. Imo, treating only commutative rings with unity is quite justified in an introductory course in abstract algebra.

• +1 A pro of groups-first is that it follows a path of "complexity" (for lack of a better word) if one sees Algebra as being based on sets and/or set theory. In Algebra, Larry C. Grove introduces Algebraic structures more or less in this order: Set -> semigroup -> monoid -> group -> ring -> ring with unity -> division ring -> field, with branches for Abelian groups, commutative rings, etc. So he is "building" structures from "simplier" entities. It seems to me that another option would be to start with vector spaces and "drill down". Jan 29, 2016 at 17:28
• @Todd Wilcox. Great idea! I wonder if anyone has written a book that "drills down" from vector spaces? It would be interesting to test this approach because Linear Algebra typically precedes Intro. Abstract Algebra in the undergrad curriculum, and students have generally struggled to learn vector spaces as their first substantive abstraction. So one could approach groups and rings by encountering things that don't admit all of the structure, requiring one to "shut off" axioms. Jan 29, 2016 at 17:49
• @JonBannon: I don't know about any books, but this course at Harvard lectured by Benedict Gross begins by introducing groups to an audience that (if I'm not mistaken) has already seen vector spaces. The course makes extensive use of Artin's Algebra (though the book itself presents groups in Ch. 2 and then vector spaces in Ch. 3). Jan 30, 2016 at 0:59
• I just finally got hold of a copy of this text. @TedShrifin is an amazing pedagogue, and it is sad that he has retired from UGA. We're very lucky his course for Math 3500 and 3510 are on youtube. I just wish he'd recorded some other courses! His text is very clearly written, beautifully constructed, and well justified in the preface. He states that he started seeing benefit from Rings first when he was teaching mostly future secondary teachers; then he started doing the same approach for regular math majors. All of the reasoning he gives is what has been stated on this page for Rings. Feb 2, 2016 at 2:38

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 algebra text. I was kind of annoyed at the time that I would (unless I wanted to fight the textbook) have to teach using a rings first approach. By the end of the semester I was convinced that rings first is the way to go.

Full disclosure: That was the only time I've taught rings first. My department (at Appalachian State University) has adopted a groups first approach (we use Gallian's Contemporary Abstract Algebra) and I respect those choices. But I still believe that a ring first approach is generally better.

Of course, "better" is subjective and highly depends on the type of course you're teaching and who your target audience is. Generally, intro to modern algebra courses (in my experience) contain a mixed bag of math and math education majors. Math education majors tend to have difficulties with the groups first approach. They see no value in the course content -- with good reason. If you start with rings, the course sells itself. Applications are immediate.

Pro: For rings first is you have immediate access to a large class of examples (number rings, polynomial rings, matrix rings, etc). With groups first you must spend a large amount of time building examples before you can really do anything.

Pro: Immediate applications. There are many easily accessible applications that are appealing to students. For groups your applications tend to be more obscure geometry/combinatorics or cryptography (where arguably it's not applying group theory as much as number theory). On the other hand, for example, with rings you can immediately get into the theory of factoring polynomials and related things that will seem quite appealing to secondary education majors.

Con: A ring's axiom system is longer and more complicated. Honestly, I don't really accept this as a serious con. We teach vector spaces in linear algebra before groups -- even more complicated! The fact that a ring is a group plus ... is typically not exploited in introductory courses. As an example, do you see undergraduate texts mention that the distributive laws are merely statements that multiplication operators are group homomorphisms (under addition) and thus since the identity maps to the identity we have $r0=0$ for all $r$? I think not.

Pro: Rings first, familiar examples are less weird. Every time I teach (groups first) I spend more time that I'd like reminding confused students that $\mathbb{R}$ (and other number rings) is a group under addition not multiplication ("rule of thumb -- does it contain zero?"). When you tell a student that $\mathbb{Z}$ is a group under addition, the immediate question that comes to mind is "What about multiplication?"

I rather enjoyed teaching out of undergraduate Hungerford. He presents the integers and basic number theory first. Then modular arithmetic. Next, he presents polynomials -- you get the division and Euclidean algorithms again. Then quotients of polynomial rings. Now with two very concrete classes of examples in hand (with a lot of concrete calculations behind you) you do abstract ring theory and hit quotients again. By the time you cover abstract quotient rings, you've seen quotients in the context of integers and polynomials so that the abstraction seems like a natural idea. Finally, group theory is unrolled as the next step of abstraction.

I felt that this approach not only benefitted the math majors in the audience (there was plenty of time to deal with abstract systems and write abstract proofs) but it also carried along a large group of students that normally get turned off before you get to define a kernel and homomorphism.

Now if you're just teaching talented math majors -- groups first is more efficient and if they can't handle it -- they should change majors (cough cough). I prefer rings first for a general audience. Groups first for honors/pure math people.

• Same here. See my summary: matheducators.stackexchange.com/questions/228/…
– user173
Feb 3, 2016 at 10:22
• I agree, we used the Hungerford text in my undergraduate degree and it was one of my favorite courses. He makes an explicit pitch in the Preface: "Although I have no doubts that doing rings before groups is best for most students, it took me almost a decade to accept this conclusion. So I am quite sympathetic with those who are reluctant to abandon the conventional 'groups first' order. I ask only that they give this approach a fair classroom trial: 'Try it, you'll like it.'". Feb 21, 2016 at 22:00

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 mathematics teachers taking the course (like there are at my place).

The main "pro" is pedagogical: Rings are more familiar objects to students and, for pre service teachers, much closer to the math that they will be teaching in middle and high schools than groups are. The "everyday" objects we use in math -- integers, polynomials, matrices -- are all instances of rings. That also makes them groups of course, if you strip away multiplication, but it's more natural to work with these objects on their own terms without stripping away operations.

The idea is that students should work with what is familiar to them and work outward toward abstraction, rather than face abstraction first and then see familiar things as instances of those abstract classes. I would say that for most students I've seen, this is definitely the case and especially for future teachers.

The main "con" is that once you learn some abstract algebra with familiar objects it can be hard to make the jump to abstraction -- students tend to want to map integer-like operations and properties onto everything because we started with objects that behave like the integers. Nowhere is this more apparent than when moving from rings to groups, where students have a hard time in my experience grasping the idea of having only one operation to work with.

For audiences that are mostly not future teachers but future professional mathematicians, or maybe computer scientists, this is something to keep in mind -- those students need to learn to handle abstraction up front and doing groups first (or even semigroups? Or categories?) might be better.

Disclaimer: I work in the same department as Hodge, Schlicker, and Sundstrom and I am pretty partial to their book.

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 some people have already pointed out, there are much more familiar examples to work with.

I actually see this as the reason to work with groups first. Getting students familiar with the type of abstractions they will see in algebra and then further along in algebraic topology is a good thing. Some of the structures we create are incredibly unintuitive and it took me a long time before I could really feel like I was okay with the abstraction taking place in algebra. Starting with groups forced me to learn to become more comfortable with these abstract structures. Moving on to rings can then be a step where things actually become easier (instead of more difficult if you're going rings -> groups) because the structures become more "natural".