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Signature-based algorithms are the latest and most efficient approach as oftoday to compute Gr\"obner bases for polynomial systems over fields. Recently,possible extensions of these techniques to general rings have attracted theattention of several authors.  In this paper, we present a signature-based version of M\"oller's classicalvariant of Buchberger's algorithm for computing strong Gr\"obner bases overPrincipal Ideal Domains (or PIDs). It ensures that the signatures do notdecrease during the algorithm, which makes it possible to apply classicalsignature criteria for further optimization. In particular, with the F5criterion, the signature version of M\"oller's algorithm computes a Gr\"obnerbasis without reductions to zero for a polynomial system given by a regularsequence. We also show how Buchberger's chain criterion can be implemented soas to be compatible with the signatures.  We prove correctness and termination of the algorithm. Furthermore, we havewritten a toy implementation in Magma, allowing us to quantitatively comparethe efficiency of the various criteria for eliminating S-pairs.

Signature-based M\"oller's algorithm for strong Gr\"obner bases over PIDs