  We study a class of $\mathbb{Z}$-graded algebras introduced by Bell and
Rogalski. Their construction generalizes in large part that of rank one
generalized Weyl algebras (GWAs). We establish certain ring-theoretic
properties of these algebras and study their connection to GWAs. We classify
the simple weight modules in the infinite orbit case and provide a partial
classification in the case of orbits of finite order.
