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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]] |