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AbstractLet (ℌ,μ,α){({\mathfrak{H}},\mu,\alpha)} be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field 𝔽{{\mathbb{F}}}.A basis ℬ={ei}i∈I{{\mathcal{B}}=\{e_{i}\}_{i\in I}} of ℌ{{\mathfrak{H}}} is called multiplicative if for any i,j∈I{i,j\in I}, we have that μ(ei,ej)∈𝔽ek{\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and α(ei)∈𝔽ep{\alpha(e_{i})\in{\mathbb{F}}e_{p}} for some k,p∈I{k,p\in I}.We show that if ℌ{{\mathfrak{H}}} admits a multiplicative basis, then it decomposes as the direct sum ℌ=⊕rℑr{{\mathfrak{H}}=\bigoplus_{r}{{\mathfrak{I}}}_{r}} of well-described ideals admitting each one a multiplicative basis.Also, the minimality of ℌ{{\mathfrak{H}}} is characterized in terms of the multiplicative basis and it is shown that, in case ℬ{{\mathcal{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.
Georgian Mathematical Journal – de Gruyter
Published: Aug 1, 2021
Keywords: Hom-algebra; multiplicative basis; infinite dimensional algebra; structure; 17A60; 17A30
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