Regular Hom-algebras admitting a multiplicative basis

Abstract

Abstract Let ( 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 = { e i } 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 ⁢ ( e i , e j ) ∈ 𝔽 ⁢ e k {\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and α ⁢ ( e i ) ∈ 𝔽 ⁢ e p {\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 = ⊕ r ℑ r {{\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.