===== Left module =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $M,R$ |

| @#FFBB00: definiendum | @#FFBB00: $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{leftModule}(M,R)$ |

| @#55CCEE: context     | @#55CCEE: $\mathcal M\in \mathrm{abelianGroup}(M)$ |
| @#55CCEE: context     | @#55CCEE: $\mathcal R\in \mathrm{ring}(R)$ |
| @#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.

| $x,y\in M$ |
| $r,s\in R$ |

| @#55EE55: postulate   | @#55EE55: $r*(x+y) = (r*x)+(r*y)$ |
| @#55EE55: postulate   | @#55EE55: $(r\ \hat+\ s)* x = (r* x)+(s* x)$ |
| @#55EE55: postulate   | @#55EE55: $(r\ \hat*\ s)* x = r* (s* x)$ |

==== 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.

One generally speaks of an $R$-left-module over $M$. Here $R$ and $M$ are just sets.
=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Module_%28mathematics%29|Module]]
==== Parents ====
=== Context ===
[[Ring]]