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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.