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Direct decompositions of acd groups with primary regulating index. (English) Zbl 0869.20038
Arnold, David M. (ed.) et al., Abelian groups and modules. Proceedings of the international conference at Colorado Springs, CO, USA, August 7–12, 1995. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 182, 233-241 (1996).
The authors prove the following significant theorem. Theorem 3.5. Let $$G$$ be an almost completely decomposable group whose regulating index is a power of $$p$$. Write $$G=\bigoplus_{i=1}^nG_i$$ with indecomposable summands $$G_i$$. Then each $$G_i$$ has a local “near-endomorphism ring” $$\mathbb{Z}_p\otimes\text{End }G_i$$. If $$G$$ is nearly isomorphic to a sum $$\bigoplus_{j=1}^mG_j'$$ for indecomposable almost completely decomposable groups $$G_j'$$, then $$m=n$$ and after relabeling $$G_i$$ is nearly isomorphic to $$G_i'$$ for every $$i$$.
This theorem demonstrates that the common and well-known pathological direct decompositions of almost completely decomposable groups are due entirely to the presence and interaction of different primes of the regulating index. It is an important step toward an understanding of the possible decompositions of general almost completely decomposable groups.
The proof is ring-theoretical making use of the (Schultz) near-endomorphism ring $$\mathbb{Z}_p\times\text{End }G$$. The authors show that this ring is semi-perfect and that idempotents modulo its Jacobson radical lift to idempotents of the ring. An Azumaya-Krull-Schmidt Theorem then asserts unique decompositions in the near-isomorphism category. Finally, the important theorem of Arnold saying that near-decompositions imply decompositions with direct summands nearly isomorphic to the near-summands, does the rest.
For the entire collection see [Zbl 0853.00035].