# 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. 1.

$(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$ for all $\mathbf{u},\mathbf{v},\mathbf{w}\in M$

2. 2.

$\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$ for all $\mathbf{u},\mathbf{v}\in M$

3. 3.

There exists an element $\mathbf{0}\in M$ such that $\mathbf{u}+\mathbf{0}=\mathbf{u}$ for all $\mathbf{u}\in M$

4. 4.

For any $\mathbf{u}\in M$, there exists an element $\mathbf{v}\in M$ such that $\mathbf{u}+\mathbf{v}=\mathbf{0}$

5. 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. 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. 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