Generalized Weyl algebra ($D(\sigma ,t)$)

For a ring $D$, an automorphism $\sigma$ and central $t$, adjoin $X,Y$ with $Xd=\sigma (d)X$, $Yd={\sigma }^{-1}(d)Y$, $YX=t$ and $XY=\sigma (t)$. This construction includes Weyl algebras and rank-one universal enveloping algebras.

For a commutative Noetherian base $D$ of finite dimension $d$, the noncommutative Krull dimension of $D(\sigma ,t)$ equals $d$ unless a height-$d$ maximal ideal is periodic under $\sigma$ or contains infinitely many translates ${\sigma }^n(t)$. This is the commutative-base specialization of Theorem 5.3 in Krull dimension of Generalized Weyl Algebras with non-commutative coefficients.