Abstract
In this paper we consider the Krull-Schmidt-Grothendieck ring K(Q) for certain full subcategories Q of the category of quasihomomorphisms of finite rank torsion free modules over a Dedekind domain W. More specifically, we will be concerned with modules G such that W’ @ G is a Butler W’-module for some suitable finite integral extension W’ of W. We will usually also ‘assume that the typeset of W’ @ G consists of idempotent types. Let K = K(@‘) and let N = nil rad K. By a strongly indecomposable domain in Q, we mean a W-algebra D which is a commutative integral domain and such that the underlying W-module of D is strongly indecomposable and belongs to Q. If D is such a domain and G is any torsion free W-module, we let D-rank G = rank, Hom(G, D). We set P, = {[Cl [H] E K ] D-rank G = D-rank H}. We let E be the subring of K generated by all [D] such that D is a strongly indecomposable domain in 5F. Consider the following four properties:
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