Module

Definition

A module over a Ring $R$ is an algebraic structure that generalizes the notion of Vector Space by allowing scalars from a ring instead of a field.

A left $R$-module $M$ consists of:

• An abelian group $(M,+)$
• A scalar multiplication operation $\cdot :R\times M\to M$

satisfying the following axioms for all $r,s\in R$ and $m,n\in M$:

• $r\cdot (m+n)=r\cdot m+r\cdot n$
• $(r+s)\cdot m=r\cdot m+s\cdot m$
• $(rs)\cdot m=r\cdot (s\cdot m)$
• $1_R\cdot m=m$ (if $R$ has a multiplicative identity)

Right Modules

A right $R$-module is defined similarly, with scalar multiplication $M\times R\to M$ satisfying analogous axioms with the order reversed.

Examples

• Every Vector Space is a module over a Field
• Every Abelian Group is a $\mathbb{Z}$-module
• Every ring $R$ is a module over itself
• The zero module $\{0\}$ is a module over any ring

Module Homomorphisms

A module homomorphism $f:M\to N$ between $R$-modules is a group homomorphism that respects scalar multiplication:

$f(r\cdot m)=r\cdot f(m)$

for all $r\in R$ and $m\in M$.

Category of Modules

The category of left $R$-modules, denoted $R\text{-}\mathbf{Mod}$ or ${\mathbf{Mod}}_R$, has:

• Objects: left $R$-modules
• Morphisms: module homomorphisms

This category is abelian and has all limits and colimits.

Related Concepts

• Vector Space: Modules over a field
• Ring: The structure providing scalars
• Abelian Group: The underlying additive structure
• Module Homomorphism: Morphisms between modules
• Tensor Product: A construction combining modules
• Category: Modules form a category