Normal Quadratics in Ore Extensions,Quantum Planes,and Quantized Weyl Algebras

Let R be a Noetherian domain and let (σ,δ) be a quasi-derivation of R such that σ is an automorphism. There is an induced quasi-derivation on the classical quotient ring Q of R. Suppose F=t ²−v is normal in the Ore extension Rt;σ,δ] where v∈R. We show F is prime in Rt;σ,δ] if and only if F is irreducible in Qt;σ,δ] if and only if there does not exist w∈Q such that v=σ(w)w−δ(w). We apply this result to classify prime quadratic forms in quantum planes and quantized Weyl algebras.