We determine simplicity criteria in characteristics 0 and p for a ubiquitous class of iterated skew polynomial rings in two indeterminates over a base ring. One obstruction to simplicity is the possible existence of a canonical normal element z. In the case where this element exists we give simplicity criteria for the rings obtained by inverting z and the rings obtained by factoring out the ideal generated by z. The results are illustrated by numerous examples including higher quantized Weyl algebras and generalizations of some low-dimensional symplectic reflection algebras.

Rings of invariants are identified for some automorphisms 9 of certain iterated skew polynomial rings R, including the enveloping algebra of sl 2 (k), the Weyl algebra A, and their quantizations. We investigate how finite-dimensional simple R-modules split over the ring of invariants R s and how finite-dimensional simple R 9 -modules extend to R.

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