  Given a topological ring $R$, we study semitopological $R$-modules, construct
their completions, Bohr and borno modifications. For every topological space
$X$, we construct the free (semi)topological $R$-module over $X$ and prove that
for a $k$-space $X$ its free semitopological $R$-module is a topological
$R$-module. Also we construct a Tychonoff space $X$ whose free semitopological
$R$-module is not a topological $R$-module.
