# Rings in parallel with groups in abstract algebra

In a previous question, I asked about the pros and cons of teaching rings before groups in abstract algebra. Recently, it has come to my attention that there is a third approach - a unified approach - as exemplified by Matej Brešar in his Undergraduate Algebra: A Unified Approach in which he introduces the various algebraic structures in parallel. See https://www.maa.org/press/maa-reviews/undergraduate-algebra-a-unified-approach for a review by Louisa Catalano, from which I quote:

The key novelty and unique feature of the book is that, in this first part, analogous topics on different algebraic structures are considered simultaneously (for example, the section on substructures introduces subgroups, subrings, subfields, etc., subsequently; or another example, the section on normal subgroups and quotient groups is followed by the section on ideals and quotient rings).

What are the pros and cons of taking a parallel approach to algebraic structures instead of teaching them sequentially?

I think this is an interesting question. In the US undergraduate mathematics curriculum, one often finds a sequence of courses "Abstract Algebra I" and "Abstract Algebra II." I think there is lots of variation. Typically groups are in the first course, but Sylow theorems might be in the second course, along with rings. And the second course is often "topical" depending on the instructor's mathematical interests and perspectives.

As I think about this question of teaching rings in parallel with groups, I wonder how this would interface with a two-semester sequence Abstract I/II. At many institutions, only the first course is required for majors, especially for secondary education mathematics majors. The second course is then offered as a elective. Therefore, I think the question must be considered in terms of the local audience and requirements for the major and its various emphases.

Another factor, which is not too significant in this case, is consideration of transfer-credit equivalency. By the time a student is taking abstract algebra, they are probably settled into the institution, the department, and emphasis for their mathematics major and so they will not be transferring in or out. But if this question were transplanted to teaching Calculus in an innovative approach--say introduce integration very early in Calculus 1 (Apostol's classic textbook starts with integration!), then caution is needed. At the lower division level, transferring from one institution to another is common and so "transfer-credit currency" becomes an important consideration if we strive to be transfer student friendly.

Group theory has lots of applications in the sciences, such as to crystallography and quantum mechanics. Rings and fields basically don't, except in trivial ways like doing linear algebra over the complex numbers rather than the reals. I assume that's the reason why the curriculum is normally arranged the way it is, so that STEM students have the option of taking only a first-semester algebra class that focuses on groups.

I double-majored in math and physics as an undergrad, then went on to become a physicist. I never used the material from my second-semester algebra class. My first-semester algebra class ended up by developing some of the theory of the classification of finite groups, which from what I remember was intellectual stimulating but had no applications for me. From my point of view, that time would have been better spent on introducing Lie groups and representations.

• That's a good point, although I might add that rings and fields do have applications to cryptography and error correcting codes, among other subjects. STEM students in computer science or electrical engineering might benefit from learning rings and fields with the aforementioned topics in mind.
– J W
Feb 27 '21 at 15:33
• So somehow physicists use operators on Hilbert space without knowing about rings? Feb 27 '21 at 15:35
• Building on the comment of @GeraldEdgar, physicists use convolution in digital image and signal processing so $*$ gives us a ring and algebra structure on integrable functions. So I do not think it is fair to say that applications of rings in physics are basically trivial. Feb 27 '21 at 16:23
• @GeraldEdgar: Yes, physicists use operators on Hilbert spaces without knowing rings.
– user507
Feb 28 '21 at 0:58

I think this certainly makes a lot more sense than teaching rings before groups. I think it might be a wash whether you teach all of the parts of groups and then all of the parts of rings and note the similarities along the way or if you collate them like this book does, but at least this seems pedagogically viable.

I think my larger concern is how the definition of these structures are spiraled on top of each other. I've seen definitions of a group as a set endowed with a binary operation that satisfies these seven axioms and a ring is a set endowed with two binary operations that satisfies twelve axioms (where the number of axioms might vary). I think it's much stronger to start by defining a monoid as a structure and then say that a group is a monoid where each element has an inverse and a ring is a commutative group with respect to addition and a monoid with respect to multiplication that satisfies the distributive properties. If you do that, then notions like subrings and ideals will make more sense to the students whether you teach groups and rings separately or concurrently.

• Bresar does start briefly with semigroups and monoids, as it turns out.
– J W
Feb 26 '21 at 16:45
• Yeah, just based on calling it a unified approach in the title would be an encouraging sign to me. Having read the review, I'm interested in getting the book for myself. It's not Dover Press, alas, but \$45 doesn't seem unreasonable for a good intro abstract algebra book. Feb 26 '21 at 16:47
• Correction to my comment above: the author's name is actually Bre&scaron;ar, in which I have used an HTML code for the diacritic caron over the "s". This displays correctly in my question, but not in this comment.
– J W
Feb 26 '21 at 17:33