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.
