Left module


context $M,R$
definiendum $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{leftModule}(M,R)$
context $\mathcal M\in \mathrm{abelianGroup}(M)$
context $\mathcal R\in \mathrm{ring}(R)$
context $*:R\times M\to M$

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.

$x,y\in M$
$r,s\in R$
postulate $r*(x+y) = (r*x)+(r*y)$
postulate $(r\ \hat+\ s)* x = (r* x)+(s* x)$
postulate $(r\ \hat*\ s)* x = r* (s* x)$


“$*$” 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.


Wikipedia: Module



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