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module [2013/08/07 14:19] nikolaj |
module [2015/12/11 19:48] nikolaj |
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| ===== Module ===== | ===== Module ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $M,R$ | | + | | @#55CCEE: context | @#55CCEE: $M,R$ | |
| - | | @#55EE55: $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{module}(\mathcal M,\mathcal R)$ | | + | | @#55EE55: postulate | @#55EE55: $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{module}(\mathcal M,\mathcal R)$ | |
| - | | @#88DDEE: $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{leftModule}(\mathcal M,\mathcal R)$ | | + | | @#55CCEE: context | @#55CCEE: $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{leftModule}(\mathcal M,\mathcal R)$ | |
| - | | @#88DDEE: $\mathcal M\in \mathrm{abelianGroup}(M)$ | | + | | @#55CCEE: context | @#55CCEE: $\mathcal M\in \mathrm{abelianGroup}(M)$ | |
| Now denote the multiplication in the ring $\mathcal R$ by "$\ \hat*\ $". | Now denote the multiplication in the ring $\mathcal R$ by "$\ \hat*\ $". | ||
| Line 12: | Line 12: | ||
| | $r,s\in R$ | | | $r,s\in R$ | | ||
| - | | @#55EE55: $r*s=s*r$ | | + | | @#55EE55: postulate | @#55EE55: $r*s=s*r$ | |
| - | ==== Ramifications ==== | + | ==== Discussion ==== |
| - | === Discussion === | + | |
| A module is a left module with a //commutative// ring acting on the group. | A module is a left module with a //commutative// ring acting on the group. | ||
| One generally speaks of an $R$-module over $M$. Here $R$ and $M$ are just sets. | One generally speaks of an $R$-module over $M$. Here $R$ and $M$ are just sets. | ||
| - | ==== Reference ==== | + | |
| + | {{https://www.youtube.com/watch?v=yZ2dO6Fy5Kc}} | ||
| + | |||
| + | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Module_%28mathematics%29|Module]] | Wikipedia: [[http://en.wikipedia.org/wiki/Module_%28mathematics%29|Module]] | ||
| - | ==== Context ==== | + | ==== Parents ==== |
| === Subset of === | === Subset of === | ||
| [[Left module]] | [[Left module]] | ||
