On ideal and subalgebra coefficients in semigroup algebras

Let k[S] be a semigroup algebra with coefficients in a commutative field k, and let U be a one-sided ideal in k[S] or a k-subalgebra of k[S], It is proven that there exists a smallest subfield k′ ≤ k such that U as a one-sided ideal resp. as a k-algebra can be generated by elements in k′[S]. By means of an example it is shown that the straightforward extension of this result to finitely generated commutative k-algebras is not valid.