Classification of Simple Modules of the Ore Extension $$K[X][Y; f\frac{d}{dX}]$$

For the algebras $$\Lambda $$Λ in the title of the paper, a classification of simple modules is given, an explicit description of the prime and completely prime spectra is obtained, the global and the Krull dimensions of $$\Lambda $$Λ are computed.

Down–up algebras over a polynomial base ring 𝕂[t1,…,tn]

We study a class of down–up algebras 𝒜(α, β, ϕ) defined over a polynomial base ring 𝕂[t1,…,tn] and establish several analogous results. We first construct a 𝕂-basis for the algebra 𝒜(α, β, ϕ). As an application, we completely determine the center of 𝒜(α, β, ϕ) when char 𝕂 = 0, and prove that the Gelfand–Kirillov dimension of 𝒜(α, β, ϕ) is n + 3. Then, we prove that 𝒜(α, β, ϕ) is a noetherian domain if and only if β ≠ 0, and 𝒜(α, β, ϕ) is Auslander-regular when β ≠ 0. We show that the global dimension of 𝒜(α, β, ϕ) is n + 3, and 𝒜(α, β, ϕ) is a prime ring except when α = β = ϕ = 0. Finally, we obtain some results on the Krull dimensions, isomorphisms and automorphisms of 𝒜(α, β, ϕ).