What is module? — Mathematics

Definition 3.1 University Mathematics — Year 3 · Chapter 3 — Modules over a Principal Ideal Domain

An $A$ -module is an abelian group $(M, +)$ with a scalar multiplication $A \times M \to M$ satisfying the vector-space axioms: $a(x + y) = ax + ay$ , $(a + b)x = ax + bx$ , $(ab)x = a(bx)$ , $1x = x$ . Submodules, quotients $M/N$ , morphisms ( $A$ -linear maps), direct sums $\bigoplus_i M_i$ , and the isomorphism theorems are defined and proved word for word as for vector spaces and abelian groups; in particular $M/\ker f \cong \operatorname{im} f$ for a morphism $f$ .

Examples

Example 3.2

The three motivating cases.

$A = K$ a field: modules are vector spaces.
$A = \Z$ : modules are exactly abelian groups ( $nx$ is forced to be $x + \dots + x$ ), submodules are subgroups.
$A = K[X]$ : a module is a $K$ -vector space $V$ together with the $K$ -linear map $u\colon x \mapsto X\cdot x$ — conversely, every pair $(V, u)$ with $u \in \mathcal L(V)$ becomes a $K[X]$ -module by $P \cdot x = P(u)(x)$ . The submodules are precisely the $u$ -stable subspaces.

An ideal of $A$ is exactly a submodule of $A$ ; a quotient ring $A/I$ is an $A$ -module. Unlike vector spaces, modules can have torsion: in $\Z/6\Z$ , the element $\bar 3 \ne 0$ is killed by $2 \neq 0$ .

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