  We study the structure of certain modules $V$ over linear spaces $W$ with
restrictions neither on the dimensions nor on the base field $\mathbb F$. A
basis $\mathfrak B = \{v_i\}_{i\in I}$ of $V$ is called multiplicative respect
to the basis $\mathfrak B' = \{w_j\}_{j \in J}$ of $W$ if for any $i \in I, j
\in J$ we have either $v_iw_j = 0$ or $0 \neq v_iw_j \in \mathbb Fv_k$ for some
$k \in I$. We show that if $V$ admits a multiplicative basis then it decomposes
as the direct sum $V=\bigoplus_k V_k$ of well-described submodules admitting
each one a multiplicative basis. Also the minimality of $V$ is characterized in
terms of the multiplicative basis and it is shown that the above direct sum is
by means of the family of its minimal submodules, admitting each one a
multiplicative basis.
