Modules over linear spaces admitting a multiplicative basis

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 ${\mathfrak{B}}^{\mathrm{\prime }}=\{w_j{\}}_{j\in J}$ of $W$ if for any $i\in I,j\in J$ we have either $v_i{w}_j=0$ or $0\mathrm{\ne }{v}_i{w}_j\in \mathbb{F}{v}_k$ for some $k\in I$. We show that if $V$ admits a multiplicative basis then it decomposes as the direct sum $V={⨁}_k{V}_k$ 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.