  Signature-based algorithms have become a standard approach for Gr\"obner
basis computations for polynomial systems over fields, but how to extend these
techniques to coefficients in general rings is not yet as well understood.
  In this paper, we present a proof-of-concept signature-based algorithm for
computing Gr\"obner bases over commutative integral domains. It is adapted from
a general version of M\"oller's algorithm (1988) which considers reductions by
multiple polynomials at each step. This algorithm performs reductions with
non-decreasing signatures, and in particular, signature drops do not occur.
When the coefficients are from a principal ideal domain (e.g. the ring of
integers or the ring of univariate polynomials over a field), we prove
correctness and termination of the algorithm, and we show how to use signature
properties to implement classic signature-based criteria to eliminate some
redundant reductions. In particular, if the input is a regular sequence, the
algorithm operates without any reduction to 0.
  We have written a toy implementation of the algorithm in Magma. Early
experimental results suggest that the algorithm might even be correct and
terminate in a more general setting, for polynomials over a unique
factorization domain (e.g. the ring of multivariate polynomials over a field or
a PID).
