  This paper deals with the notion of Gr\"obner $\delta$-base for some rings of
linear differential operators by adapting the works of W. Trinks, A. Assi, M.
Insa and F. Pauer. We compare this notion with the one of Gr\"obner base for
such rings. As an application, and following a previous work of A. Assi, we
give some results on finiteness and on flatness of finitely generated left
modules over these rings.
