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left_module [2013/08/06 13:32] nikolaj |
left_module [2013/08/07 14:19] nikolaj |
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===== Left module ===== | ===== Left module ===== | ||
==== Definition ==== | ==== Definition ==== | ||
- | | @#88DDEE: $\mathcal M\in \mathrm{abelianGroup}(M)$ | | + | | @#88DDEE: $M,R$ | |
- | | @#88DDEE: $\mathcal R\in \mathrm{ring}(R)$ | | + | |
- | | @#55EE55: $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{leftModule}(\mathcal M,\mathcal R)$ | | + | | @#55EE55: $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{leftModule}(M,R)$ | |
+ | | @#88DDEE: $\mathcal M\in \mathrm{abelianGroup}(M)$ | | ||
+ | | @#88DDEE: $\mathcal R\in \mathrm{ring}(R)$ | | ||
| @#88DDEE: $*:R\times M\to M$ | | | @#88DDEE: $*:R\times M\to M$ | | ||
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"$*$" 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. | "$*$" 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 (also) calls $M$ the module. | + | One generally speaks of an $R$-left-module over $M$. Here $R$ and $M$ are just sets. |
==== Reference ==== | ==== Reference ==== | ||
Wikipedia: [[http://en.wikipedia.org/wiki/Module_%28mathematics%29|Module]] | Wikipedia: [[http://en.wikipedia.org/wiki/Module_%28mathematics%29|Module]] |