We report on the recent theoretical adoption and implementation of non-commutative Gröbner bases both over fields and over the ring Z with more principal ideal rings in mind, which was accomplished in parallel. In both cases we are based on the Letterplace correspondence of La Scala and Levandovskyy. The corresponding subsystem of Singular is called Letterplace. In addition to division with remainder algorithm, we also provide algorithms for syzygy bimodules and lifting. Despite a certain similarity, the case of rings as coefficients demonstrates crucial intrinsic differences, compared to the case of fields. We also remark on models of computation and on decidability.