  In this paper we present a new efficient variant to compute strong Gr\"obner
basis over quotients of principal ideal domains. We show an easy lifting
process which allows us to reduce one computation over the quotient $R/nR$ to
two computations over $R/aR$ and $R/bR$ where $n = ab$ with coprime $a, b$.
Possibly using available factorization algorithms we may thus recursively
reduce some strong Gr\"obner basis computations to Gr\"obner basis computations
over fields for prime factors of $n$, at least for squarefree $n$. Considering
now a computation over $R/nR$ we can run a standard Gr\"obner basis algorithm
pretending $R/nR$ to be field. If we discover a non-invertible leading
coefficient $c$, we use this information to try to split $n = ab$ with coprime
$a, b$. If no such $c$ is discovered, the returned Gr\"obner basis is already a
strong Gr\"obner basis for the input ideal over $R/nR$.
