Ergodipotent maps and commutativity of elements in noncommutative rings and algebras with twisted intertwining

A property of algebraic dependence between two commuting elements is shown to hold in a more general setting than that in which it has previously been established. Key conditions are identified and some methods for establishing them are given. Moreover the class of algebras with a generalised Weyl structure, generalising the so-called Generalised Weyl Algebras (GWAs) or hyperbolic rings, is introduced and studied. We also present an interesting class of algebras which are not GWAs but share many of their properties by virtue of their generalised Weyl structure. For these classes of algebras, centralisers and algebraic dependence are investigated.