Introduction to Modules

A group action lets a group $G$ permute the elements of a set. A module is the same idea one level up: a ring$R$ acts on an abelian group $M$, and the action is required to respect both the addition of $M$ and the addition and multiplication of $R$. Two familiar theories are special cases. When $R$ is a field the axioms are exactly those of a vector space; when $R=\mathbb{Z}$ they are exactly those of an abelian group. Module theory studies both at once: a single structure theorem for modules over a principal ideal domain reproduces the classification of finite abelian groups and the Jordan and rational canonical forms of a matrix from one argument.¹

The definition

The word left records that ring elements are written on the left of module elements. A right module puts them on the right, and axiom $(rs)m=r(sm)$ becomes $m(rs)=(mr)s$; when $R$ is commutative the two notions coincide and we drop the adjective. The action is a ring analogue of a group action: writing $R^{\times }$ for the units of $R$, the map $m\mapsto rm$ for a unit $r$ is a bijection of $M$, and the axioms make $R^{\times }$ act on the set $M$ by additive automorphisms.

Two boundary cases:

• $R$ a field $F$. The axioms are verbatim the vector-space axioms. Modules over a field are vector spaces over that field.
• $R=\mathbb{Z}$. For any abelian group $A$ the rule $na=a+\cdots +a$ ($n$ times, with $0a=0$ and $(-n)a=-(na)$) is the only unital action of $\mathbb{Z}$ on $A$. So $\mathbb{Z}$-modules are the same as abelian groups, and there is nothing to choose: the group structure forces the action.

Submodules and the submodule criterion

Submodules are the subobjects of the theory. Over a field they are subspaces; over $\mathbb{Z}$ they are subgroups. Every $M$ has the two trivial submodules $0$ and $M$. Testing the definition directly means checking closure under subtraction and under the action separately; a single condition does both.

First examples

• $R$ over itself. Any ring $R$ is a left module over itself, the action being ring multiplication. Its submodules are precisely the left ideals of $R$. A cyclic submodule $Ra$ is a principal (left) ideal, so a principal ideal domain is exactly a commutative domain in which every submodule of $R$ is cyclic.
• Free module of rank $n$. For $n\in {\mathbb{Z}}^{+}$, the set $R^n=\{(a_1,\dots ,a_n):a_i\in R\}$ with componentwise addition and action $r(a_1,\dots ,a_n)=(r{a}_1,\dots ,r{a}_n)$ is the free module of rank $n$. It is the ring analogue of coordinate space $F^n$.
• Abelian groups as $\mathbb{Z}$-modules. Any abelian group $A$ carries its unique $\mathbb{Z}$-action. If $A$ has an element $x$ of finite order $n$, then $nx=0$ with $n\ne 0$: unlike a vector space, a module can have a nonzero element killed by a nonzero scalar. These are torsion elements; they distinguish module theory from linear algebra.
• Killing an ideal. If a two-sided ideal $I$ annihilates $M$ (that is $am=0$ for all $a\in I$, $m\in M$), then $M$ becomes an $R/I$-module via $(r+I)m=rm$. When $I$ is a maximal ideal and $IM=0$, the module $M$ is a vector space over the field $R/I$. For an elementary abelian $p$-group $A$ with $px=0$, this makes $A$ a vector space over ${\mathbb{F}}_p=\mathbb{Z}/p\mathbb{Z}$.

Vector spaces as $F[x]$-modules

This example makes a vector space into a module over a polynomial ring. Let $F$ be a field, $V$ a vector space over $F$, and $T:V\to V$ a linear transformation. Define an action of $F[x]$ on $V$ by letting $x$ act as $T$ and extending: for $p(x)=a_n{x}^n+\cdots +a_{1}x+a_0$,

where $T^n$ is the $n$-fold composite and $T^0=I$. The module axioms hold, and the constant polynomials act as the original scalar multiplication, so this action extends the $F$-module structure to an $F[x]$-module structure. The choice of $T$ is the only freedom, so a single vector space $V$ carries many different $F[x]$-module structures — one for each linear operator on it.

Because $F[x]$ is a principal ideal domain, its module structure is tightly constrained, and that constraint becomes detailed information about $T$: its rational canonical form and Jordan canonical form.

Homomorphisms

Equivalently, a single condition $\phi (rx+y)=r\phi (x)+\phi (y)$ captures both requirements. Every module homomorphism is in particular a homomorphism of the underlying abelian groups, but not conversely: the group map $x\mapsto 2x$ on $\mathbb{Z}$ is a $\mathbb{Z}$-module homomorphism, while over $R=F[x]$ the map $f(x)\mapsto f(x^2)$ is a ring homomorphism that fails to be $F[x]$-linear. The names specialize as expected.

• Over a field, module homomorphisms are linear transformations.
• Over $\mathbb{Z}$, they are abelian-group homomorphisms — the action condition is automatic, since scalar multiplication is repeated addition.

Kernels and images are submodules, by the submodule criterion applied to $\phi$. The set ${\mathrm{Hom}}_R(M,N)$ of all homomorphisms $M\to N$ is itself an abelian group under pointwise addition, and an $R$-module when $R$ is commutative. The endomorphisms ${\mathrm{End}}_R(M)={\mathrm{Hom}}_R(M,M)$ form a ring under composition — the endomorphism ring of $M$.

Quotient modules

Every submodule is the kernel of a projection, so quotients always exist. This is simpler than the group case: a module is an abelian group, so every submodule is a normal subgroup, and there is no obstruction to forming the quotient.

The elements of $M/N$ are the cosets $x+N$, translates of the submodule $N$ partitioning $M$.

The isomorphism theorems

Because a module is an abelian group with extra structure, the four isomorphism theorems carry over verbatim from groups; one only checks that the group isomorphisms are $R$-linear.⁴

The first theorem matters most: every homomorphic image of $M$ is a quotient of $M$, so understanding $M$ up to isomorphism reduces to understanding its submodules and the quotients by them. The sum $A+B=\{a+b:a\in A,b\in B\}$ appearing in the second theorem is the smallest submodule containing both $A$ and $B$.

Reductions of this shape, specialized to $R=\mathbb{Z}$, package statements about finite abelian groups; the structure theorem over a PID turns them into a complete classification.

Fields versus principal ideal domains

The definition splits the subject cleanly. Over a field, the added ring structure degenerates and one recovers linear algebra: every module is free and dimension is a complete invariant. Over $\mathbb{Z}$ or $F[x]$ — more generally over a principal ideal domain — torsion appears, modules need not have a basis, and the finer invariants of the structure theorem are needed.