## Bewertete p-Gruppen und ein Satz von Szele.(German)Zbl 0575.20048

It is a result of Szele that a $$p^ n$$-bounded subgroup B of an abelian p-group G is a direct summand if $$B\cap p^ nG=0$$. This result is generalized to valuated groups. More precisely, if A is a finite valuated p-group then conditions are found which force any monomorphism $$\alpha$$ : $$\oplus_{I}A\to G$$ to split where G is a p-local valuated group. As replacement for the classical subgroup $$p^ nG$$ the radical $$R_ A(G)=\cap \{Ker(f):$$ $$f\in Morph(G,A)\}$$ is used. The two generalizations of Szele’s theorem can now be stated.
Corollary 4.3: Let A be a finite, indecomposable, valuated p-group and G a p-valuated group which possesses a morphism $$\phi$$ on G to a group $$\oplus_{J}A$$ with $$Ker(\phi)=R_ A(G)$$. Then every monomorphism $$\alpha$$ : $$\oplus_{I}A\to G$$ with $$Im(\alpha)\cap R_ A(G)=0$$ splits.
The morphism $$\phi$$ always exists for simply presented p-groups A and G. Theorem 6.8: The following statements are equivalent for a simply presented, finite, valuated p-group A.
(a) $$A=\oplus^{n}_{i=1}\oplus_{m_ i}S(X_ i)$$ where $$m_ i$$ is finite, the $$X_ i$$ are finite, irretractable, incomparable, valuated p- trees and S(X) denotes the simply presented valuated p-group with p-basis X.
(b) If P is a direct summand of $$\oplus_{I}A$$ then every monomorphism $$\alpha$$ on P into a simply presented valuated p-group G splits if and only if $$\alpha (P)\cap R_ A(G)=0.$$
An important tool in the proofs is a category equivalence between the category of direct sums $$\oplus_{I}A$$ of a finite valuated p-group A and the category of free Morph(A,A)-modules. Furthermore, a subgroup $$G(T^*)$$ of the valuated group G, defined for any valuated p-tree T, is discussed at some length as a possible replacement of the classical subgroup $$p^ nG$$. However, when both A and G are simply presented valuated p-groups and $$A=S(T)$$ then $$R_ A(G)=G(T^*)$$. The paper is very readable and contains a useful summary of the concepts of valuated groups, trees, simply presented etc. In Satz 6.7 read $$S(X_ i)$$-free instead of A-free.
 [1] Albrecht, U., Endomorphism rings and $$A$$-projective torsion-free abelian groups, (Abelian Group Theory, Proceedings. Abelian Group Theory, Proceedings, Honolulu 1983. Abelian Group Theory, Proceedings. Abelian Group Theory, Proceedings, Honolulu 1983, Lecture Notes in Mathematics, Vol. 1006 (1983), Springer-Verlag: Springer-Verlag New York/Berlin), 209-227 [2] Bass, H., Finistic dimension and homological generalization of semi-primary rings, Trans. Amer. Math. Soc., 95, 466-488 (1960) · Zbl 0094.02201 [3] Fuchs, L., (Infinite Abelian Groups, Vols. I and II (1970/1973), Academic Press: Academic Press New York) [5] Hunter, R.; Richman, F.; Walker, E., Simply presented valuated groups, J. Algebra, 49, 125-133 (1977) · Zbl 0383.20038 [6] Hunter, R.; Walker, E., Valuated $$p$$-groups, (Abelian Group Theory, Proceedings. Abelian Group Theory, Proceedings, Oberwohlfach 1981. Abelian Group Theory, Proceedings. Abelian Group Theory, Proceedings, Oberwohlfach 1981, Lecture Notes in Mathematics, Vol. 874 (1981), Springer-Verlag: Springer-Verlag New York/Berlin), 350-372 · Zbl 0518.20045 [7] Popescu, N., Abelian Categories with Applications to Rings and Modules (1973), Academic Press: Academic Press New York · Zbl 0271.18006 [8] Richman, F.; Walker, E., Valuated groups, J. Algebra, 56, 145-167 (1979) · Zbl 0401.20049