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" -> center dot center dot center dot -> A(alpha) -> center dot center dot center dot, is attached where a is an arbitrary ordinal. In general, all rings A(a) are distinct. A new construction of rings, the inner (sigma, tau, a)-extension of a ring, is introduced (where sigma and tau are endomorphisms of a ring D and a is an element of D).