WILD TORSION MODULES OVER WEYL ALGEBRAS AND GENERAL TORSION MODULES OVER HNPS

Let A(1) be the first Weyl algebra over a field F of characteristic ze ro. 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 genera ted) of socle-height 2 is wild in a very strong sense. Among the appli cations, we show: (i) Essentially any F-algebra can occur as the endom orphism algebra of an A(1)-module of socle-height 2; (ii) A(1) has ver y 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 t he basic theory of finitely generated modules over general HNPs, by sh owing that every such module is a direct sum of right ideals and homom orphic images of right ideals, and by proving a simultaneous decomposi tion theorem for an arbitrary projective module and submodule. (C) 199 5 Academic Press, Inc.