Power-substitution and cancellation in the class of Abelian groups (Q1063694)

A right R-module N has the power-substitution property if for any decomposition \(M=N_ 1\oplus B_ 1=N_ 2\oplus B_ 2\) where \(N_ 1\cong N\cong N_ 2\) there is an \(n\in {\mathbb{N}}\) such that \(\oplus_{n}B_ 1\) and \(\oplus_{n}B_ 2\) have a common complement in \(\oplus_{n}M\). If \(n=1\) we obtain the substitution property. These properties can be introduced in an arbitrary category. A ring E has the right power-substitution property if the right E-module E has it. A ring E has unity in the stable range if \(a_ 1b_ 1+a_ 2b_ 2=1\), \(a_ i,b_ i\in E\) implies that there is \(b\in E:\) \(a_ 1+a_ 2b\) is invertible. For a subfunctor T of the identity in \(Mod_ R\) such that T(B) is completely characteristic in B and T preserves finite sums the author introduces the category Walk(T) generalizing the one defined by Walker for the torsion subfunctor of the identitiy, ''neglecting'' the images of T. Canonic functors between \(Mod_ R\) and Walk(T) are introduced and used to prove: Lemma 7. If for \(A\in Mod_ R\) the ring E(T(A)) has unity in the stable range and the ring E(A)/U(A) where \(U(A)=\{x\in E(A)|\) xA\(\subseteq T(A)\}\) has the right power-substitution property then A has the power- substitution property in \(Mod_ R\). - Lemma 8. If for \(A\in Mod_ R\) the rings E(T(A)) and E(A)/U(A) have the unity in the stable range then A has the substitution property in \(Mod_ R.\) As consequences, the author solves positively Problems 6 and 7 of \textit{R. B. Warfield} jun. [in Lect. Notes Math. 616, 1-38 (1977; Zbl 0368.20032)] proving Theorem. A countble Abelian group A with finite torsion-free rank such that T(A) has finite Ulm invariants has the power-substitution property. Moreover, if A is almost divisible then A has the substitution property. Remarks. The translation is several times wrong: ''torsion-free of finite rank'' must be read ''of finite torsion-free rank''.