Classifying modules over particular rings. – Can we find a border-line in

terms of endomorphism rings?

Rüdiger Göbel

June 4, 1998

From Paul Eklof [3] (’s abstract) we learn how decomposition properties of modules in the sense of Kaplansky-Corner are related with particular rings often - but not only - constructed in the 60ties and then realized as endomorphism rings. Eklof will also consider the connection of prediction principles in V = L with what we called ”Shelah’s Black Box in [1], see Shelah [14]. This gives us the advantage to concentrate on more specific topics, some of them dealt with very recently. Nevertheless I will illustrate the use of the Black Box giving one and only one educational example:

(1) Find a (large) cotorsion-free R-module M with

(∗) End (M) = R.

We may be very special and choose R = ℤ.

I will take this opportunity to make clear that Shelah’s Black Box is really an easy ”Kinder-garden counting argument” which everybody can understand - however it is a very clever one, which turns out to be useful in many places in algebra., see [5, 2] for instance. After the most desired application as in (1) are well-understood, it seems fascinating to see how we can put the module class under additional strains and still get realization theorem as in (∗). We will concentrate on two obviously opposite main directions investigated quite recently:

(2) How free can a module M as above be? — see [6].

(3) How close to pure-injectivity [16] can we choose M with (∗) — see [7, 8] Keywords are almost free modules on one side and splitter or stones on the other

side, cf. [11, 12, 13]. Other applications are related to questions concerning the Kronecker module, see e.g. [9, 10]

References

[1] A.L.S. Corner, R. Göbel, Prescribing endomorphism algebras - A unified treatment, Proceed. London Math. Soc. (3) 50 (1985), 471 – 483.

[2] M. Dugas, R. Göbel, Solution of Philip Hall’s problem on the existence of complete locally finite p-groups and results on E.C. groups, Journal of Algebra 159 (1993) 115 - 138.

[3] P. Eklof, Modules with strange decomposition properties, Abstract for Bielefeld 1998

[4] P. Eklof, A. Mekler Almost free modules, Set-theoretic methods, North-Holland, Amsterdam 1990

[5] R. Göbel, A. Paras , Outer automorphism groups of metabelian groups, submitted to Journal of Pure and Applied Algebra 1997

[6] R. Göbel, S. Shelah, Indecomposable almost free moduoles - the local case, to appear in Canadian Journal of Math. 1998

[7] R. Göbel, S. Shelah, Cotorsion theories and splitters, submitted to Transactions AMS.

[8] R. Göbel, S. Shelah, ℵ_{1}−free splitters of cardinality ℵ_{1}, in preparation 1998

[9] R. Göbel, D. Simson, Rigid families and endomorphism algebras of Kronecker modules, (hopefully) to appear in Israel Journal of Math.

[10] R. Göbel, J. Trlifaj, Cotilting and a hierarchy of almost cotorsion groups, submitted to Journal of Algebra

[11] O. Kerner, Elementary stones, Comm. Algebra, 22 (1994) 1797 – 1806.

[12] C. M. Ringel, The braid group action on the set of exceptional sequences of a hereditary artin algebra, pp. 339 – 352 in Abelian group theory and related topics, Contemporary Math. 171 American Math. Soc., Providence, R.I. 1994

[13] C. M. Ringel, Bricks in hereditary length categories, Resultate der Mathematik 6 (1983) 64 – 70.

[14] S. Shelah, A combinatorial theorem and endomorphism rings of abelian groups II, pp. 37 – 86, in Abelian groups and modules, CISM Courses and Lectures, 287, Springer, Wien 1984 [15] T. Wakamatsu, On modules with trivial self extension, Journal of Algebra 114

(1988), 106–114.

[16] R. Warfield, Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969) 699 – 719.

Rüdiger Göbel Fachbereich 6, Mathematik und Informatik Universität Essen, 45117 Essen, Germany

e–mail: R.Goebel@Uni-Essen.De