Abstract Algebra (9): Modules — Generalizing Vector Spaces

Fri, 17 Sep 2021 09:00:00 +0000

In every linear algebra course, you learn to work over a field: real numbers, complex numbers, or perhaps a finite field. The resulting theory is remarkably clean — every subspace has a complement, every finitely generated vector space has a basis, and all bases have the same cardinality. But what happens when we replace the field with a ring?

The answer is modules: the natural generalization of vector spaces, where scalars come from a ring rather than a field. The theory is richer, the pathologies more interesting, and — perhaps most importantly — modules turn out to encompass an enormous range of mathematical objects: abelian groups (modules over $\mathbb{Z}$ ), vector spaces with a linear endomorphism (modules over $$K[x]$$ ), ideals (modules over a ring), and group representations (modules over a group ring). What initially feels like a technical generalization is actually a unifying framework that organizes much of algebra.

Modules on Chen Kai Blog

Abstract Algebra (9): Modules — Generalizing Vector Spaces