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