Direct factors of polynomial rings over finite fields

Let G_q denote the multiplicative semigroup of all monic polynomials in one indeterminate over a finite field F_q with q elements. By a direct factor of G_q is understood a subset B₁ of G_q such that, for some subset B₂ of G_q, every polynomial w G_q has a unique factorization in the form w = b₁b₂ for b_i B_i. An asymptotic formula B₁^#(n) c₁qⁿ as n → ∞ is derived for the total number B₁^#(n) of polynomials of degree n in an arbitrary direct factor B₁ of G_q, c₁ a constant depending on B₁.