  Many rings and algebras arising in quantum mechanics can be interpreted as
skew PBW (Poincar\'e-Birkhoff-Witt) extensions. Indeed, Weyl algebras,
enveloping algebras of finite-dimensional Lie algebras (and its quantization),
Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum
matrices, among many others, are examples of skew PBW extensions. In this short
paper we study the d-Hermite condition about stably free modules for skew PBW
extensions. For this purpose, we estimate the stable rank of these
non-commutative rings. In addition, and close related with these questions, we
will prove the Kronecker's theorem about the radical of finitely generated
ideals for some particular types of skew PBW extensions.

Rings and Algebras