In this paper, we give an algorithm that computes syzygies over Laurent polynomials ring:  with Ris a valuation ring (Ra valuation domain of any Krull dimension) which neither relies on Noetherianity nor on Krull dimension. The idea is based  on the isomorphism   and the syzygy computation over the polynomials ring   At the end of this paper, we prove that this algorithm is also valid for Prüfer domain.

Laurent polynomial rings, syzygies, valuation ring, constructive mathematics, Prüfer ring, computer algebra.