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