Module
$(R, M, \cdot)$: left $R$-module.
$R$: ring.
$M$: group.
$\cdot$: function from $R \times M$ to $M$ written as $r \cdot m = rm$.
$m, m_1, m_2$: elements of $M$.
$r, r_1, r_2$: elements of $R$.
$1m = m$: identity of $R$ maps
$r(m_1 + m_2) = rm_1 + rm_2$: operations in $M$ align with operations in $R$.
$(r_1 + r_2)m = r_1m + r_2m$: operations in $R$ align with operations in $M$.
$r_1(r_2m) = (r_1 r_2)m$: operations carry over.
Right $R$-module: $\cdot$ $M \times R$ to $M$.
Vector spaces
$R$: a field $F$.