Wild torsion modules over Weyl algebras and general torsion modules over HNPs Original Research Article

Let A1 be the first Weyl algebra over a field F of characteristic zero. We show that the category of A1-modules of finite length is wild, despite the fact that every A1-module of finite length is cyclic. In fact, the category of A1-modules (not necessarily finitely generated) of socle-height 2 is wild in a very strong sense. Among the applications, we show: (i) Essentially any F-algebra can occur as the endomorphism algebra of an A1-module of socle-height 2; (ii) A1 has very large indecomposable modules of finite length; (iii) There is an HNP (hereditary Noetherian prime ring) that has indecomposable modules of finite length requiring arbitrary many generators. We also complete the basic theory of finitely generated modules over general HNPs, by showing that every such module is a direct sum of right ideals and homomorphic images of right ideals, and by proving a simultaneous decomposition theorem for an arbitrary projective module and submodule.