Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
left_module [2013/08/07 14:19] nikolaj |
left_module [2014/03/21 11:11] (current) |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ===== Left module ===== | ===== Left module ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $M,R$ | | + | | @#55CCEE: context | @#55CCEE: $M,R$ | |
| - | | @#55EE55: $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{leftModule}(M,R)$ | | + | | @#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 ==== |
| - | === Parents === | + | === Context === |
| [[Ring]] | [[Ring]] | ||
