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left_module [2013/08/31 21:25]
nikolaj
left_module [2014/03/21 11:11]
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-===== Left module ===== 
-==== Definition ==== 
-| @#88DDEE: $M,R$ | 
  
-| @#FFBB00: $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{leftModule}(M,​R)$ | 
- 
-| @#88DDEE: $\mathcal M\in \mathrm{abelianGroup}(M)$ | 
-| @#88DDEE: $\mathcal R\in \mathrm{ring}(R)$ | 
-| @#88DDEE: $*:R\times M\to M$ | 
- 
-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. 
- 
-| $x,y\in M$ | 
-| $r,s\in R$ | 
- 
-| @#55EE55: $r*(x+y) = (r*x)+(r*y)$ | 
-| @#55EE55: $(r\ \hat+\ s)* x = (r* x)+(s* x)$ | 
-| @#55EE55: $(r\ \hat*\ s)* x = r* (s* x)$ | 
- 
-==== 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: [[http://​en.wikipedia.org/​wiki/​Module_%28mathematics%29|Module]] 
-==== Context ==== 
-=== Requirements === 
-[[Ring]] 
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