For a polynomial with zero constant term, a semiprime K-algebra R is called faithful f-free if every nonzero ideal of R does not satisfy f. We prove that a semiprime algebra has an essential ideal which is the direct sum of its largest faithful f-free ideal and its largest ideal satisfying the identity f. Here, faithful f-free algebras are characterized, and applications to some interesting differential identities are discussed. Especially with f = S2n (n ≥ 1), a description of semiprime rings is obtained in terms of faithful S2n-free rings and semiprime PI-rings of degree 2n. Semiprime PI-rings of degree 2n are realized through faithful S2n-rings. Finally, faithful S2n-rings are characterized.