Modules and Vector Spaces

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 $F$.