  Signature-based algorithms have brought large improvements in the
performances of Gr\"obner bases algorithms for polynomial systems over fields.
Furthermore, they yield additional data which can be used, for example, to
compute the module of syzygies of an ideal or to compute coefficients in terms
of the input generators.
  In this paper, we examine two variants of Buchberger's algorithm to compute
Gr\"obner bases over principal ideal domains, with the addition of signatures.
The first one is adapted from Kandri-Rody and Kapur's algorithm, whereas the
second one uses the ideas developed in the algorithms by L. Pan (1989) and D.
Lichtblau (2012). The differences in constructions between the algorithms
entail differences in the operations which are compatible with the signatures,
and in the criteria which can be used to discard elements.
  We prove that both algorithms are correct and discuss their relative
performances in a prototype implementation in Magma.
