examples of modules
This entry is a of examples of modules over rings. Unless otherwise specified in the example, $M$ will be a module over a ring $R$.

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Any abelian group^{} is a module over the ring of integers^{}, with action defined by $n\cdot g$ for $g\in G$ given by $n\cdot g={\sum}_{i=1}^{n}g$.

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If $R$ is a subring of a ring $S$, then $S$ is an $R$module, with action given by multiplication^{} in $S$.

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If $R$ is any ring, then any (left) ideal $I$ of $R$ is a (left) $R$module, with action given by the multiplication in $R$.

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Let $R=\mathbb{Z}$ and let $E=\{2k\mid k\in \mathbb{Z}\}$. Then $E$ is a module over the ring $\mathbb{Z}$ of integers. Further, define the sets $B=E\times E$ and $C=E\times \{0\}$ and $D=\{0\}\times E$. Then $B$, $C$, and $D$ are modules over $\mathbb{Z}\times \mathbb{Z}$, with action given by $a\cdot x=(a\cdot {x}_{1},a\cdot {x}_{2})$ if $x=({x}_{1},{x}_{2})$ even if the product is redefined as $a\cdot {x}_{1}=0$ and $a\cdot {x}_{2}=0$, but now the identity element^{} is $(1,1)$. However by our new product definition $a\cdot x=(a\cdot {x}_{1},a\cdot {x}_{2})=(0,0)$ even if $a=(1,1)$, the ring identity element originally In the more general definition of module which does not require an identity element $\mathrm{\U0001d7cf}$ in the ring and does not require $\mathrm{\U0001d7cf}\cdot m=m$ for all $m\in M$, we observe that $\mathrm{\U0001d7cf}\cdot m\ne m$ in this example just constructed. (one of the purposes of this comment is to show that all modules need not be unital ones).

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YetterDrinfel’d module. (http://planetmath.org/QuantumDouble)
Title  examples of modules 

Canonical name  ExamplesOfModules 
Date of creation  20130322 14:36:28 
Last modified on  20130322 14:36:28 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  16 
Author  mathcam (2727) 
Entry type  Example 
Classification  msc 1600 
Classification  msc 2000 
Classification  msc 1300 
Related topic  QuantumDouble 
Related topic  Module 