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 $R$.

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

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

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

• 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 $\mathrm{𝟏}$ in the ring and does not require $\mathrm{𝟏}\cdot m=m$ for all $m\in M$, we observe that $\mathrm{𝟏}\cdot m\ne m$ in this example just constructed. (one of the purposes of this comment is to show that all modules need not be unital ones).

• Yetter-Drinfel’d module. (http://planetmath.org/QuantumDouble)