ABSTRACT
In this chapter we answer completely and, as will be seen later (Section 11.9), in an computationally effective fashion the question whether an almost completely de composable group has a non-zero completely decomposable direct summand (The orem 7.3.8, Theorem 11.9.5). This supplements the known criteria for the case of almost completely decomposable groups with cyclic regulating quotient (Proposi tion 6.3.11, Proposition 6.3.10, Theorem 6.4.1). Moreover, the exact rank of a maxi mal r-homogeneous (completely decomposable) direct summand of X is determined (Theorem 7.3.16). We also establish an algorithm for deciding whether an almost completely decomposable group is completely decomposable (Theorem 7.3.18). These criteria are sometimes helpful in showing that a group is directly indecomposable since indecomposable groups must above all lack completely decomposable sum mands. Further motivation is provided by the Main Decomposition X = X cd ® X ci (Corollary 7.3.17, Theorem 9.2.7). It is also noteworthy that block-rigid crq-groups have decompositions into indecomposable summands that are unique if they are clipped. In contrast, block-rigid crq-groups that are not clipped may have a large number of essentially different direct decompositions with indecomposable sum mands (Theorem 13.1.3, Example 13.1.12, Theorem 13.1.13). Most of the results of this chapter appeared in [MN99].
