  Let $I_1\subset I_2\subset\dots$ be an increasing sequence of ideals of the
ring $\Bbb Z[X]$, $X=(x_1,\dots,x_n)$ and let $I$ be their union. We propose an
algorithm to compute the Gr\"obner base of $I$ under the assumption that the
Gr\"obner bases of the ideal $\Bbb Q I$ of the ring $\Bbb Q[X]$ and the the
ideals $I\otimes(\Bbb Z/m\Bbb Z)$ of the rings $(\Bbb Z/m\Bbb Z)[X]$ are known.
  Such an algorithmic problem arises, for example, in the construction of
Markov and semi-Markov traces on cubic Hecke algebras.
