Many transcendental division algebras from suitably conditioned iterated skew polynomial rings are nonaffine over k, and those of Weyl algebras and quantum affine spaces are affine over their centers exactly when finite-dimensional over those centers.
Osofsky,On twisted polynomial rings, J
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`New' examples of skew fields not finitely generated as algebras
Many transcendental division algebras from suitably conditioned iterated skew polynomial rings are nonaffine over k, and those of Weyl algebras and quantum affine spaces are affine over their centers exactly when finite-dimensional over those centers.