Left Module

Overview

Modules extend vector spaces by allowing scalars from a ring. This small change has major consequences: a nonzero scalar may fail to be invertible, nonzero vectors may be killed by scalars, and a basis need not exist. Modules provide one language for abelian groups, linear operators, and many structures arising from rings.

Prerequisites

Know rings with identity and abelian groups. A module uses the ring's multiplication to compose scalar actions. When the ring is noncommutative, distinguish left multiplication from right multiplication; the order cannot be discarded.

Definition and conventions

Let $R$ be a unital ring. A left $R$-module is an abelian group $M$ with an action $R\times M\to M$ satisfying

All variables range over their indicated structures. A submodule is an additive subgroup closed under the scalar action. A module homomorphism preserves addition and that action.

For a right module, scalars are written on the right and the associativity law is $(mr)s=m(rs)$. Over a commutative ring a left module gives a right module by $mr=rm$, but this conversion is not automatic for general rings.

Intuition

The ring supplies allowable operations on the additive group. Unlike a field, it may contain scalars that compress or annihilate information. Consequently modules retain divisibility and torsion phenomena that vector spaces erase.

Examples and non-examples

Every abelian group is a $\mathbb Z$-module: positive integers act by repeated addition, negative integers by additive inverses. For $M=\mathbb Z/6\mathbb Z$, the nonzero class of $2$ is killed by the scalar $3$. This cannot happen for a nonzero vector scaled by a nonzero field element.

The same module is not free over $\mathbb Z$. Any nonzero element of a free abelian group has infinite additive order, whereas every element here has finite order.

A vector space $V$ with an endomorphism $T$ becomes an $F[x]$-module by setting $p(x)v=p(T)v$. The equation $(pq)(T)=p(T)q(T)$ verifies the scalar associativity axiom. This example makes operator theory part of module theory.

A nonabelian group cannot be a module with its existing group operation as addition, since the additive group of a module must be abelian.

Important properties

The quotient $M/N$ by a submodule inherits a scalar action, because changing a representative by an element of $N$ changes its multiple by another element of $N$. For a homomorphism $f:M\to P$, the kernel and image are submodules and

The map sends the class of $m$ to $f(m)$; the kernel relation proves well-definedness and injectivity.

Free modules have bases, but general modules are described by generators and relations. Finite generation and freeness are distinct conditions.

Connections and further reading

Tensor products combine module actions, and exact sequences express relations between modules. Read Clark, Section 2, particularly Some Constructions and Free Modules, for these structural constructions.