Let $(ℌ,\mu ,\alpha )$ be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field $𝔽$. A basis $ℬ={\left\{e_i\right\}}_{i\in I}$ of $ℌ$ is called multiplicative if for any $i,j\in I$, we have that $\mu (e_i,e_j)\in 𝔽e_k$ and $\alpha \left(e_i\right)\in 𝔽e_p$ for some $k,p\in I$. We show that if $ℌ$ admits a multiplicative basis, then it decomposes as the direct sum $ℌ={\oplus }_r{\Im }_r$ of well-described ideals admitting each one a multiplicative basis. Also, the minimality of $ℌ$ is characterized in terms of the multiplicative basis and it is shown that, in case $ℬ$, in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division.

中文翻译：

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接受乘法的正则Hom代数

让 $（ℌ，\mu ，\alpha ）$ 是任意维度且在任意基域上的规则Hom代数 $𝔽$。基础$ℬ={\left\{{Ë}_{\mathrm{一世}}\right\}}_{\mathrm{一世}\in \mathrm{一世}}$ 的 $ℌ$ 被称为乘法 $\mathrm{一世}，Ĵ\in \mathrm{一世}$，我们有 $\mu （{Ë}_{\mathrm{一世}}，{Ë}_{Ĵ}）\in 𝔽{Ë}_{ķ}$ 和 $\alpha （{Ë}_{\mathrm{一世}}）\in 𝔽{Ë}_p$ 对于一些 $ķ，p\in \mathrm{一世}$。我们证明如果$ℌ$ 接受一个乘法基础，然后分解为直接和 $ℌ={\oplus }_{\mathrm{[R}}{\Im }_{\mathrm{[R}}$公认的理想中的每一个都以乘法为基础。还有，最小的$ℌ$ 根据乘法基础来表征，并且表明 $ℬ$，此外，它是除法的基础，然后上述直接和是由其最小理想的族组成的，每个理想都承认除法的乘法基础。