Iterated skew polynomial rings of Krull dimension two (Q1312362)

Let \(A\) be a finitely generated algebra over an algebraically closed field \(k\). Let \(\alpha\) be a \(k\)-automorphism and let \(R\) denote the iterated skew polynomial ring \(A[y; \alpha][x; \alpha^{-1}, \delta]\). The main intent of this article is to show that if \(A\) is \(\alpha\)-simple and a principal ideal domain and if \(X\) is a finitely generated right \(R\)-module such that both \(X_ x\) and \(X_ y\) have finite length, then \(X\) has finite length. This eliminates the need in earlier work of the author to assume that all finite dimensional modules were semisimple.