  Many rings and algebras arising in quantum mechanics, algebraic analysis, and
non-commutative algebraic geometry can be interpreted as skew PBW
(Poincar\'e-Birkhoff-Witt) extensions. In the present paper we study two
aspects of these non-commutative rings: its finitely generated projective
modules from a matrix-constructive approach, and the construction of the
Gr\"obner theory for its left ideals and modules. These two topics could be
interesting in future eventual applications of skew $PBW$ extensions in
functional linear systems and in non-commutative algebraic geometry.
