Within the quantum function algebra Fq[GLn], we study the subset Fq[GLn] — introduced in [Ga1] — of all elements of Fq[GLn] which are Z [ q, q ] –valued when paired with Uq(gln) , the unrestricted Z [ q, q ] –integral form of Uq(gln) introduced by De Concini, Kac and Procesi. In particular we yield a presentation of it by generators and relations, and a PBW-like theorem. Moreover, we give a direct proof that Fq [GLn] is a Hopf subalgebra of Fq[GLn], and that Fq[GLn] ∣∣ q=1 ∼= UZ(gln ∗) . We describe explicitly its specializations at roots of 1, say ε, and the associated quantum Frobenius (epi)morphism from Fε[GLn] to F1[GLn] ∼= UZ(gln ∗) , also introduced in [Ga1]. The same analysis is done for Fq [SLn] and (as key step) for Fq [Matn] .