Inner generalized Weyl algebras and their simplicity criteria

The aim of the paper is to introduce a new class of rings — the inner generalized Weyl algebras (IGWA) — and to give simplicity criteria for them. For each IGWA A a derivative series of IGWAs, A→A′→A′′→⋯→A(α)→⋯, is attached where α is an arbitrary ordinal. In general, all rings A(α) are distinct. A new construction of rings, the inner(σ,τ,a)-extension of a ring, is introduced (where σ and τ are endomorphisms of a ring D and a∈D).