Math. Z. 223, 421 -434 (1996) Mathematische Zeitschr © Springer-Verlag 1996 Elementary modules Otto Kerner, Frank Lukas Mathematisches Institut, Heinrich-Heine-Universit~t, Universit~tsstrasse 1, D-40225 Diissel- dorf, Germany Received 16 July 1992; in final form 27 February 1995 Introduction Let A be a finite dimensional, connected wild hereditary k-algebra, k an algebraically closed field. We denote by A-reg the full subcategory of regu- lar A-modules in A-mod. This category is closed under images and extensions, but, contrary to the tame situation, not closed under kernels and cokernels. A nonzero regular module E is called elementary, if there is no nontrivial regular submodule X, such that E/X is regular, too. Since the Auslander-Reiten trans- lations t and z- define an equivalence on A-reg, a module E is elementary, if and only if tiE is elementary, for all integers i. It follows from the definition that each nonzero regular module X has a filtration 0 -- X0 C Xl C ... C Xr = X with elementary composition fac- tors X,./X,._], hence the class ~ of elementary modules is the smallest class of regular modules, whose extension-closure is A-reg. By definition the elemen- tary modules are exactly the quasi-simple regular modules, if the algebra A is tame. We will show in part 2 that - parallel to the tame situation - there exist only finitely many Coxeter-orbits of dimension-vectors of elementary modules. Totally different to the tame case is, that a z-sincere module E is elemen- tary only if dimk Ext(E,E) > 2 holds (Theorem 3.4) and that there exist infinitely many algebras whose elementary modules all are stones, that is in- decomposable modules without self-extensions, see part 4. In this ease The- orem 2.1 then says that there are only finitely many z-orbits of elementary modules. Finally we show the occurrence of elementary modules in natural con- structions: If B is (wild) concealed, if M is a regular B-modules then the one-point extension B[M] is a tilted algebra only if M is elementary. Simi- larly, if the quiver ~ is a wild star with vertices {0 ..... n} where 0 denotes the center of the star, if M is an indecomposable k~-module with dim M =