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.
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机译:我们研究了线性空间$ \ mathbb {W} $上某些$ k $模块$ \ mathbb {V} $的结构,但对$ \ mathbb {V} $和$ \ mathbb {W} $的尺寸均没有限制也不在$ \ mathbb F $的基本字段上。 $ \ mathbb {V} $的基数$ \ mathfrak B = \ {v_i \} _ {i \ in I} $相对于基础$ \ mathfrak B'= \ {w_j \} _ {j \如果$ \ sigma \ inS_n,$ $ i_1,\ dots,i_k \ in I $和$ j_ {k + 1},\ dots,j_n \ in J $我们有$ [v_ {i_1},\点,v_ {i_k},w_ {j_ {k + 1}},\点,w_ {j_n}] _ {\ sigma} \ in \ mathbb {F} v_ {r_ {\ sigma}} $代表$ r _ {\ sigma} \ in I $。我们表明,如果$ \ mathbb {V} $接受乘法基础,则它将分解为直接描述的$ k $ -submodules $ \ mathbb {V} = \ bigoplus _ {\ alpha} V _ {\ alpha} $ V _ {\ alpha} $每个人都承认有乘法基础。 $ \ mathbb {V} $的极小值也以乘法为基础,并且表明上述直接和是通过其minimum $ k $子模块族来实现的,每个子模块都允许以乘法为基础。最后,我们在具有乘法基础的任意$ n $ -ary代数上研究了具有乘法基础的$ k $-模块的应用。
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