Wild torsion modules over Weyl algebras and general torsion modules over HNPs

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Let A 1 be the first Weyl algebra over a field F of characteristic zero. We show that the category of A 1 -modules of finite length is wild, despite the fact that every A 1 -module of finite length is cyclic. In fact, the category of A 1 -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 A 1 -module of socle-height 2; (ii) A 1 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.