module

Let $R$ be a ring with identity. A left module $M$ over $R$ is a set with two binary operations, $+:M\\times M\\longrightarrow M$ and $\\cdot:R\\times M\\longrightarrow M$ , such that

1.
$(\\mathbf{u}+\\mathbf{v})+\\mathbf{w}=\\mathbf{u}+(\\mathbf{v}+\\mathbf{w})$ for all $\\mathbf{u},\\mathbf{v},\\mathbf{w}\\in M$
2.
$\\mathbf{u}+\\mathbf{v}=\\mathbf{v}+\\mathbf{u}$ for all $\\mathbf{u},\\mathbf{v}\\in M$
3.
There exists an element $\\mathbf{0}\\in M$ such that $\\mathbf{u}+\\mathbf{0}=\\mathbf{u}$ for all $\\mathbf{u}\\in M$
4.
For any $\\mathbf{u}\\in M$ , there exists an element $\\mathbf{v}\\in M$ such that $\\mathbf{u}+\\mathbf{v}=\\mathbf{0}$
5.
$a\\cdot(b\\cdot\\mathbf{u})=(a\\cdot b)\\cdot\\mathbf{u}$ for all $a,b\\in R$ and $\\mathbf{u}\\in M$
6.
$a\\cdot(\\mathbf{u}+\\mathbf{v})=(a\\cdot\\mathbf{u})+(a\\cdot\\mathbf{v})$ for all $a\\in R$ and $\\mathbf{u},\\mathbf{v}\\in M$
7.
$(a+b)\\cdot\\mathbf{u}=(a\\cdot\\mathbf{u})+(b\\cdot\\mathbf{u})$ for all $a,b\\in R$ and $\\mathbf{u}\\in M$

A left module $M$ over $R$ is called unitary or unital if $1_{R}\\cdot\\mathbf{u}=\\mathbf{u}$ for all $\\mathbf{u}\\in M$ .

A (unitary or unital) right module is defined analogously, except that the function $\\cdot$ goes from $M\\times R$ to $M$ and the scalar multiplication operations act on the right. If $R$ is commutative, there is an equivalence of categories between the category of left $R$ –modules and the category of right $R$ –modules.

Title	module
Canonical name	Module
Date of creation	2013-03-22 11:49:14
Last modified on	2013-03-22 11:49:14
Owner	djao (24)
Last modified by	djao (24)
Numerical id	11
Author	djao (24)
Entry type	Definition
Classification	msc 13-00
Classification	msc 16-00
Classification	msc 20-00
Classification	msc 44A20
Classification	msc 33E20
Classification	msc 30D15
Synonym	left module
Synonym	right module
Related topic	MaximalIdeal
Related topic	VectorSpace