Injective Homogeneity and Homological Homogeneity of the Ore Extensions

In this paper we prove that under some natural conditions, the Ore extensions and skew Laurent polynomial rings are injectively homogeneous or homologically homogeneous if so are their coefficient rings. Specifically, we prove that if R is a commutative Noetherian ring of positive characteristic, then A.(R), the n.. Weyl algebra over R, is injectively homogeneous (resp. homologically homogeneous) if R has finite injective dimension (resp. global dimension).