$k$-Modules over linear spaces by $n$-linear maps admitting a multiplicative basis

We study the structure of certain $k$-modules $\mathbb{V}$ over linear spaces $\mathbb{W}$ with restrictions neither on the dimensions of $\mathbb{V}$ and $\mathbb{W}$ nor on the base field $\mathbb{F}$. A basis $\mathfrak B = {v_i}{i\in I}$of$$\mathbb{V}$$ is called multiplicative with respect to the basis ${\mathfrak{B}}^{\mathrm{\prime }}=\{w_j{\}}_{j\in J}$ of $\mathbb{W}$ if for any $\sigma \in S_n,$ $i_1,\dots ,i_k\in I$ and $j_{k+1},\dots ,j_n\in J$ we have $[v{i_1},\dots, v_{i_k}, w_{j_{k+1}}, \dots, w_{j_n}]{\sigma} \in $\mathbb{F}$v{r_{\sigma}}$forsome$r_{\sigma} \in I$\mathrm{.}Weshowthatif$$\mathbb{V}$$admitsamultiplicativebasisthenitdecomposesasthedirectsum$$\mathbb{V}$ = \bigoplus_{\alpha} V_{\alpha}$ofwelldescribed$k$-submodules$V_{\alpha}$eachoneadmittingamultiplicativebasis\mathrm{.}Alsotheminimalityof$$\mathbb{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 $k$-submodules, admitting each one a multiplicative basis. Finally we study an application of $k$-modules with a multiplicative basis over an arbitrary $n$-ary algebra with multiplicative basis.