Summary: Let F be a field, \(R\supset B\) be F-algebras such that for every element \(x\in R\) there is a polynomial f(t) with f(x)\(\in B\). The following results are proved: If R is prime and has no nonzero algebraic one-sided ideals, then B is prime with no nonzero algebraic one-sided ideals; if moreover R is noncommutative, then the algebras of quotients of R and B coincide. In that case R is a right Goldie algebra iff B is such an algebra.