On ideal and subalgebra coefficients in a class <i>k</i>-algebras

Abstract

Let k be a commutative field with prime field $k₀$ and A a k- algebra. Moreover, assume that there is a k-vector space basis $ω$ of A that satisfies the following condition: for all $ω₁, ω₂ ∈ ω$ ,the product $ω₁ω₂$ is contained in the $k₀$-vector space spanned by $ω$. It is proven that the concept of minimal field of definition from polynomial rings and semigroup algebras can be generalized to the above class of (not necessarily associative) k-algebras. Namely, let U be a one-sided ideal in A or a k-subalgebra of A. Then there exists a smallest $k' ≤ k$ with U-as one-sided ideal resp. as k-algebra—being generated by elements in the $k'$-vector space spanned by $ω$.