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Module

Definition

A module over a ring $R$ is an abelian group $(M,+)$ with a multiplication action $R \times M \to M$ written $(r,m) \mapsto r.m$ where:

• $1.m=m \forall m \in M$ (where $1 \in R$ is the multiplicative identity).
• $r_1.(r_2.m)=r_1 r_2.m \forall r_1,r_2 \in R; m \in M$
• $(r_1+r_2).m=r_1.m+r_2.m$ and $r.(m_1+m_2).m=r.m_1+r.m_2 \forall r,r_1,r_2 \in R; m,m_1,m_2 \in M$

In very non-standard language that I just made up, the action is quasi-associative and is doubly quasi-distributive, and the action of the identity element is the identity transformation.

ACHTUNG! Above is the definition of a left-module, since the action is on the left. Right-modules are defined analogously but I'm only going to talk about left-modules here.

Properties

Each element of $R$ defines a transformation on $M$.

The annihilator of an element $m \in M$ is the set $Ann_R (m) \subseteq R$ of elements $r \in R$ where $r.m=0$. The annihilator is a left ideal of $R$. If $Ann_R(m) \ne \{0\}$ we say $m$ is a torsion element; these are somewhat analogous to zero-divisors in rings.

A linearly independent set is a subset $S \subset M$ where, whenever $r_1 s_1 + r_2 s_2 + ... + r_k s_k$ for some $r_i \in R$ and distinct $s_i \in S$, we have $r_1=r_2=...=r_k=0$.

A spanning set, or generating set, is a set $S \subseteq M$ that generates $M$; specifically, any $m \in M$ can be expressed as a linear combination $m=r_1 s_1 + ... + r_k s_k$ for some $r_i \in R, s_i \in S$.

A basis is a linearly independent spanning set. If a basis exists, we say $M$ is free.

Types

If every element in $M$ is a torsion element, we say it is a torsion module.

If $M$ contains no torsion elements, we say it is torsion-free. If $R$ is a principal ideal domain then $M$ is also free.

If $M$ has a basis, we say it is free. If $R$ is an integral domain then $M$ is also torsion-free.

Examples

The trivial module, that contains only the zero element.

Submodule: if an abelian subgroup $N$ of $M$ is such that $r.N \subseteq N$ for all $r \in R$, it is a submodule. For any submodule we can define a quotient module $M/N$.

Any ring can be considered as a module over itself, with the action equal to left-multiplication. Any left-ideal can be considered as a submodule of the ring. Any quotient ring can be considered as a quotient module of the ring.

A ring is an integral domain if and only if it is a torsion-free module over itself.

See also

Group action - using group elements to define transformations.

all these pages adapted with probably insufficient credit from my university's lecture notes