In this paper we present first steps in using signature-based Gröbner basis algorithms like Faugère's F5 or GVW for computation over Euclidean rings. We present problems appearing when having to deal with coefficients and zero divisors and give practical solution techniques. A hybrid algorithm is presented trying to combine the advantages of signature-based and non-signature-based Gröbner basis computation. For some examples speedups are achieved due to faster finding good reducers with the hybrid technique.