Modules

Modules “naturally” arise when considering several different algebraic objects that share the concept of a ring action:

1. The classic example is a \(k\)-vector space \(V\). Scalar multiplication can be thought of as the action of a well-behaved ring (i.e. field) on an abelian group \(V\).
2. More generally, any abelian group is naturally acted upon by \(\mathbb{Z}\).
3. Any ring, of course, naturally acts on itself. Note that a ring is also an abelian group by forgetting about the ring multiplication!

Accordingly, a module is a ring action on an abelian group compatible with the group operation. Of course, as anyone in algebra knows, a variety of important examples arise in both the commutative and non-commutative cases. Personally, I find presenting the commutative case first more intuitive because the more immediate examples (at least to me) are commutative (the integers, the integers modulo some integer, vector spaces, polynomials with coefficients from a commutative ring, etc.)

Modules over commutative rings

• Matrix Canonical Forms
  • Jordan Canonical Form
  • Rational Canonical Form

Modules over non-commutative rings

TODO