MODULES WITH STRANGE DECOMPOSITION PROPERTIES

PAUL EKLOF

We will consider classes of abelian groups and modules where the Kaplansky test problems have negative answers. Kaplansky [9, pp. 12f] posed two test problems in order to “know when we have a satisfactory [structure] theorem. ... We suggest that a tangible criterion be employed: the success of the alleged structure theorem in solving an explicit problem.” The two problems were:

(I) If A is isomorphic to a direct summand of B and conversely, are A and B isomorphic?

(II) If A ⊕ A and B ⊕ B are isomorphic, are A and B isomorphic?

Of particular interest is the method developed by Corner ([2], [3]) which, by realizing certain rings as endomorphism rings (sometimes modulo an ideal), provides negative answers to both test problems (for a given class) as special cases of an even more extreme pathology. More precisely, Corner’s method — where applicable — yields, for any positive integer r, a module M_{r} (in the class) such that for any positive integers m and k, the direct sum of m copies of M_{r} is isomorphic to the direct sum of k copies of M_{r} if and only if m is congruent to k mod r. Then we obtain negative answers to both test problems by letting A = M_{2} (≅ M_{2}⊕M_{2}⊕M_{2}) and B = M_{2} ⊕ M_{2}. We shall refer to the existence of such a module M_{r} (for each r) as the “Corner pathology”.

For a number of classes of modules, the Corner pathology has been achieved with the help of “prediction principles”, either the “diamond principles” that are a consequence of V = L, the Axiom of Constructibility, or some form of the “Black Box”: a prediction principle, due to Shelah, which is provable in ordinary set theory, ZFC: see [5].

Shelah [11] has derived the Corner pathology for the class of all modules over any ring which is not pure semisimple. As in the following, this is a consequence of a very weak form of realization result.

A subring A of End(M) is said to be algebraically closed in End(M) when every finite set of ring equations with parameters from A (i.e., polynomials in several variables over A) which is satisfied in End(M) is also satisfied in A. For each r there is a ring A_{r} considered by Leavitt, Cohn and Corner (cf. [10],[1],[4]) which is universal for the property that any R − A_{r}-bimodule M is isomorphic to M^{r}. This ring has the property that if A_{r} is algebraically closed in End(M) then for any positive integers m and k, M^{m} is isomorphic to M^{k} only if m is congruent to k mod r.

In [7] we consider the class of ℵ_{1}-separable abelian groups (of cardinality ℵ_{1}). An abelian group M is called ℵ_{1}-separable if every countable subset is contained in a countable free direct summand of M. Although some structural properties of ℵ_{1}-separable groups are independent of ZFC (cf. [6, Chap. VIII]) we prove in ZFC the Corner pathology for this class. The proof involves a direct construction of the pathological group M using a tree-like ladder system and a “countable template” which comes from the Corner example [3] for countable torsion-free groups with the help of results from [8]. The Corner pathology holds because A_{r} is algebraically closed in End(M); no stronger result realizing A_{r} as an endomorphism ring is known in this case (without assuming more than ZFC).

The construction can be extended inductively to obtain ℵ_{n}-separable abelian groups of cardinality ℵ_{n} with the Corner pathology for each n > 1.

References

[1] P. M. Cohn, Some remarks on the invariant basis property, Topology 5 (1969), 215–228.

[2] A. L. S. Corner, Every countable reduced torsion-free ring is an endomorphism ring, Proc. London Math. Soc. 13 (1963), 687–710.

[3] A. L. S. Corner, On a conjecture of Pierce concerning direct decompositions of Abelian groups, in Proceedings of the Colloquium on Abelian Groups, Tihany, Budapest (1964), 43–48.

[4] A. L. S. Corner, Additive categories and a theorem of W. G. Leavitt, Bull. AMS 75 (1969), 78–82.

[5] A. L. S. Corner and R. Göbel, Prescribing endomorphism algebras, a unified treatment, Proc. London Math. Soc (3) 50 (1985), 447–479.

[6] P. C. Eklof and A. H. Mekler, Almost Free Modules, North-Holland (1990).

[7] P. C. Eklof and S. Shelah, The Kaplansky test problems for ℵ_{1}-separable groups, Proc. AMS 126 (1998), 1901–1907.

[8] R. Göbel and B. Goldsmith, The Kaplansky test problems - an approach via radicals, J. Pure and Appl. Algebra 99 (1995), 331–344.

[9] I. Kaplansky, Infinite abelian groups, rev. ed., Univ. of Michigan Press (1969).

[10] W. G. Leavitt, The module type of a ring, Trans. AMS 103 (1962), 113-130.

[11] S. Shelah, Kaplansky test problem for R-modules, Israel J. Math. 74 (1991), 91–127.

University of California, Irvine

  E-mail address: peklof@math.uci.edu