$\mathbb{Z}$-graded simple rings

The Weyl algebra over a field $k$ of characteristic $0$ is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all $\mathbb{Z}$-graded simple rings of GK-dimension 2 and show that they are graded Morita equivalent to generalized Weyl algebras as defined by Bavula. More generally, we study $\mathbb{Z}$-graded simple rings $A$ of any dimension which have a graded quotient ring of the form $K[t, t^{-1}; σ]$ for a field $K$. Under some further hypotheses, we classify all such $A$ in terms of a new construction of simple rings which we introduce in this paper. In the important special case that $\operatorname{GKdim} A = \operatorname{tr.deg}(K/k) + 1$, we show that $K$ and $σ$ must be of a very special form. The new simple rings we define should warrant further study from the perspective of noncommutative geometry.

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