2.5 - Discovering Abstract Algebra: A Constructivist Approach to Module Theory

Summary

Introduction

While module theory is not a regular part of a typical undergraduate mathematics curriculum, the topic provides an excellent opportunity to tie together fundamental courses in group theory and linear algebra. Many individual faculty members, whole departments, and consortia have addressed a common student perception that the 10 to 12 courses forming a mathematics major are all fairly distinct from one another. The “seven into four” movement promoted by, among others, the United States Military Academy is a prime example of changing the curriculum in such a way that students see deep relationships among calculus, linear algebra, differential equations, multivariable functions, and so on. At the theoretical level, module theory allows students to use their background in axiomatic mathematics to see modules as generalizations of vector spaces, whose structure, under the right conditions, mimics the structure of finitely generated abelian groups.

Most undergraduate mathematics majors in the United States require essentially a one semester course in algebra, which typically covers the basics of groups, rings, and fields. Second courses in abstract algebra might go deeper into group theory (Sylow theorems, series, solvability), Galois theory, or cover special topics such as representation theory, simple groups, algebraic coding theory, etc. Rarely do students learn about modules until they are in graduate school.

I do not argue that teaching module theory is better than teaching Galois theory or representation theory to advanced undergraduates. I merely suggest that the topic deserves consideration at the undergraduate level.