  Birack modules are modules over an algebra Z[X] associated to a finite birack
X. In previous work, birack module structures on Z mod n were used to enhance
the birack counting invariant. In this paper, we use birack modules over
Laurent polynomial rings Z_n[q,1/q] to enhance the birack counting invariant,
defining a customized Alexander polynomial-style signature for each X-labeled
diagram; the multiset of these polynomials is an enhancement of the birack
counting invariant. We provide examples to demonstrate that the new invariant
is stronger than the unenhanced birack counting invariant and is not determined
by the generalized Alexander polynomial.
