Modules over a Ring

Abelian group plus a scalar action: vector spaces and abelian groups unified.

The idea

A module generalizes a vector space by letting the scalars come from an arbitrary ring rather than a field.

Definition (Module).

A left module over a ring $R$ (with $1$) is an abelian group $(M, +)$ together with a scalar action $R \times M \to M$ satisfying $r(m+n) = rm + rn, \quad (r+s)m = rm + sm, \quad (rs)m = r(sm), \quad 1 \cdot m = m$ for all $r, s \in R$ and $m, n \in M$.

These are the vector-space axioms, with the field of scalars replaced by $R$.

One definition now covers several subjects at once. When $R$ is a field, a module is exactly a vector space. When $R = \mathbb{Z}$, a module is exactly an abelian group: the action of $n$ on $m$ is forced to be $m$ added to itself $n$ times, so every abelian group carries one and only one $\mathbb{Z}$-module structure. When $R = k[x]$, a module is a vector space over $k$ together with a chosen linear operator, namely the action of $x$. Kernels, images, quotients and the isomorphism theorems work for modules just as they do for vector spaces and abelian groups, so each theorem about modules specializes to all of these settings at once.

Ways to work on it

• Walkthrough. Check the axioms that make a ring act on an abelian group, then recognize submodules and homomorphisms.
• Practice. Compute a scalar action in a cyclic module.
• Hardest. Apply the first isomorphism theorem for modules.

Not sure where to start? Take the ten-question placement test.