On ideal and subalgebra coefficients in a class k-algebras

Abstract

Let k be a commutative field with prime field and A a k- algebra. Moreover, assume that there is a k-vector space basis of A that satisfies the following condition: for all ,the product is contained in the -vector space spanned by . 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 with U-as one-sided ideal resp. as k-algebra—being generated by elements in the -vector space spanned by .