Let \(k\) be a field, let \(S\) be a semigroup, and let \(k[S]\) be the semigroup ring of \(S\) over \(k\). Suppose \(U\) is either a one-sided ideal of \(k[S]\) or else a \(k\)-subalgebra of \(k[S]\). It is shown that there exists a smallest subfield \(k'\) of \(k\) such that \(U\) is generated by elements of \(k'[S]\). If one takes \(S\) a free commutative semigroup, the theorem above becomes a well known fact from the theory of polynomial rings over fields. The author constructs an example showing that a semigroup ring cannot be changed to a general commutative algebra over a field.