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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 thebasis $\mathfrak B' = \{w_j\}_{j \in J}$ of $\mathbb{W}$ if for any $\sigma \inS_n,$ $i_1,\dots,i_k \in I$ and $j_{k+1},\dots, j_n \in J$ we have$[v_{i_1},\dots, v_{i_k}, w_{j_{k+1}}, \dots, w_{j_n}]_{\sigma} \in\mathbb{F}v_{r_{\sigma}}$ for some $r_{\sigma} \in I$. We show that if$\mathbb{V}$ admits a multiplicative basis then it decomposes as the direct sum$\mathbb{V} = \bigoplus_{\alpha} V_{\alpha}$ of well described $k$-submodules$V_{\alpha}$ each one admitting a multiplicative basis. Also the minimality of$\mathbb{V}$ is characterized in terms of the multiplicative basis and it isshown 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 anapplication of $k$-modules with a multiplicative basis over an arbitrary$n$-ary algebra with multiplicative basis.

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