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 left_module [2013/08/31 14:37]nikolaj left_module [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/08/31 21:25 nikolaj 2013/08/31 14:37 nikolaj 2013/08/07 14:19 nikolaj 2013/08/07 14:17 nikolaj 2013/08/06 13:32 nikolaj 2013/08/06 00:59 nikolaj 2013/08/06 00:42 nikolaj created Next revision Previous revision 2013/08/31 21:25 nikolaj 2013/08/31 14:37 nikolaj 2013/08/07 14:19 nikolaj 2013/08/07 14:17 nikolaj 2013/08/06 13:32 nikolaj 2013/08/06 00:59 nikolaj 2013/08/06 00:42 nikolaj created Line 1: Line 1: ===== 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. Line 14: Line 14: | $r,s\in R$ | | $r,s\in R$ | - | @#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: $(r\ \hat*\ s)* x = r* (s* x)$ | - ==== Ramifications ​==== + ==== Discussion ​==== - === 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. "​$*$"​ 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. 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]] - ==== Context ​==== + ==== Parents ​==== - === Requirements ​=== + === Context ​=== [[Ring]] [[Ring]]