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Left module
Definition
| $M,R$ |
| $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{leftModule}(M,R)$ |
| $\mathcal M\in \mathrm{abelianGroup}(M)$ |
| $\mathcal R\in \mathrm{ring}(R)$ |
| $*:R\times M\to M$ |
Now denote the addition in th group $\mathcal M$ by “$+$” as usual, and the addition and multiplication in the ring $\mathcal R$ by “$\hat+$” and “$\hat*$”, respectively.
| $x,y\in M$ |
| $r,s\in R$ |
| $r*(x+y) = (r*x)+(r*y)$ |
| $(r\ \hat+\ s)* x = (r* x)+(s* x)$ |
| $(r\ \hat*\ s)* x = r* (s* x)$ |
Discussion
“$*$” is an action of the ring on the group from the left. If the ring is commutative, then one need not distinguish between left- and right module.
One generally speaks of an $R$-left-module over $M$. Here $R$ and $M$ are just sets.
Reference
Wikipedia: Module
Context
Requirements
