We study the structure of certain k-modules 𝕍 over linear spaces 𝕎 with restrictions neither on the dimensions of 𝕍 and 𝕎 nor on the base field 𝔽. A basis 𝔅 = { v i } i ∈ I of 𝕍 is called multiplicative with respect to the basis 𝔅 ′ = { w j } j ∈ J of 𝕎 if for any σ ∈ S n , i 1 , … , i k ∈ I and j k + 1 , … , j n ∈ J we have [ v i 1 , … , v i k , w j k + 1 , … , w j n ] σ ∈ 𝔽 v r σ for some rσ ∈ I. We show that if 𝕍 admits a multiplicative basis then it decomposes as the direct sum 𝕍 = ⊕ α V α of well described k-submodules Vα each one admitting 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 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.