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$k$-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 $\mathfrak B' = \{w_j\}_{j \in J}$ of $\mathbb{W}$ if for any $σ\in S_n,$ $i_1,\dots,i_k \in I$ and $j_{k+1},\dots, j_n \in J$ we have $[v_{i_1},\dots, v_{i_k}, w_{j_{k+1}}, \dots, w_{j_n}]_σ \in \mathbb{F}v_{r_σ}$ for some $r_σ \in I$. We show that if $\mathbb{V}$ admits a multiplicative basis then it decomposes as the direct sum $\mathbb{V} = \bigoplus_α V_α$ of well described $k$-submodules $V_α$ each one admitting a multiplicative basis. Also the minimality of $\mathbb{V}$ 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.