This dissertation takes a close look into a Frobenius skew polynomial ring where some of typical invariants from noncommutative algebra do not provide any useful information about the ring. Yoshino provides some nice results for a general Frobenius skew polynomial ring in [9], however, there is still significant potential to study and identify more aspects of these rings. Here, we apply standard techniques from noncommutative algebra taking a finitely generated subspace and attempt to count the number of generators needed for powers of the subspace. We find that in certain cases where the base ring is the commutative polynomial ring or a semigroup ring, that a nonhomogeneous recurrence develops in the counting and an invariant arises naturally when solving this recurrence. We define this invariant as the Gelfand-Kirillov base and show examples where it arises.