  This note surveys the historical background of the Gr\"obner basis theory for
D-modules and linear rewriting theory. The objective is to present a deep
interaction of these two fields largely developed in algebra throughout the
twentieth century. We recall the work of M. Janet on the algebraic analysis on
linear partial differential systems that leads to the notion of involutive
division. We present some generalizations of the division introduced by M.
Janet and their relations with Gr\"obner basis theory.
