What exactly is an $R$-module?
"If $R$ is commutative, then left $R$-modules are the same as right $R$-modules and are simply called $R$-modules."
The definition of left $R$-module: $M$ is a left $R$-module if $M$ is an abelian group and $R$ a ring acting on $M$ such that
(i) $r(m_1 + m_2) = rm_1 + rm_2$
(ii) $(r_1 + r_2 ) m = r_1 m + r_2 m$
(iii) $1m = m$
(iv) $r_1 (r_2m) = (r_1 r_2) m$
I don't understand what commutativity of $R$ has to do with the module being left and right. If $R$ is commutative it means that $r_1 r_2 = r_2 r_1$. Now how does it follow from that that $rm = mr$? $M$ is not a subset of $R$, it could be anything so how does commutativity of $R$ make elements of $M$ and $R$ commute, too?