LetDbe an integral domain with quotient fieldK. We investigate conditions under which certain primitive polynomials are products of principal primes. Let * be a finite character star operation onD[X] (e.g., * =dort) and let * also denote the star operation induced onDbyI*= (I[X])*∩KwhereIdenotes a nonzero fractional ideal ofD. Then the following conditions are equivalent: (1)Dis integrally closed and each *-invertible *-ideal is principal, (2) IfPis a prime upper to 0 containing anf∈D[X] with, thenPis principal, and (3) Forf∈D[X] withfis a product of principal primes.