Non-Cohen–Macaulay unique factorization domains in small dimensions

We construct examples of non-Cohen–Macaulay unique factorization domains in small dimension. We find a unique factorization domain of dimension 3 which is not a Cohen–Macaulay ring. Moreover, there is an example of a five-dimensional affine ring S over a field k with the property that S is a non-Cohen–Macaulay unique factorization domain whenever Char k=2, while it is a Gorenstein non-factorial ring for Char k≠2. The arguments for the proofs are conceptional as well as based on a Computer Algebra System like Singular or Macaulay. For the theoretical background we investigate the factorial closure of the symmetric algebra of certain monomial modules.