# Definition: Module

Let $$(R, + ,\cdot)$$ be a ring with identity $$1_R$$ and let $$M=(M,+)$$ be a commutative group. We call $$_RM$$ a left module, if there is a binary operation. $\odot:\cases{R\times M\longrightarrow M,\cr(r,v)\longmapsto r\odot v,}$

with the following properties: * $$r\odot (u+v)=(r\odot u)+(r\odot v)$$, * $$(r+s)\odot u=(r\odot u)+(s\odot u)$$, * $$(rs)\odot u =r\odot (s\odot u)$$, * $$1\odot u=u$$.

for all $$r,s\in R$$ and all $$u,v\in M$$.

Analogously, we call $$M_R$$ a right module, if there is a binary operation. $\odot:\cases{M\times R\longrightarrow M,\cr(v,r)\longmapsto v\odot r,}$

with the following properties: * $$(u+v)\odot r=(u\odot r)+(v\odot r)$$, * $$u \odot (r+s)=(u\odot r)+(u\odot s)$$, * $$u \odot (rs) =u\odot (r\odot s)$$, * $$u\odot 1=u$$.

If $$R$$ is a commutative unit ring, then both definitions are equivalent and $$M$$ is called an module.

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### References

#### Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück