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On some class of Abelian groups with hereditary endomorphism rings. (Russian) Zbl 0636.20029

Necessary and sufficient conditions for a module over a torsion free ring to be flat are found (proposition 1.2). The result obtained is used for the description of endoflat groups. Let A be an abelian torsion free group of finite rank, R be the endomorphism ring E(A) of A, p be a prime. Put \(A_ p=A/pA\), \(R_ p=R/pR\). A group A is endoflat if and only if \(A\otimes {\mathbb{Q}}\) is a projective \(R\otimes {\mathbb{Q}}\)-module and for any prime p \(A_ p\) is a projective \(R_ p\)-module (corollary 1.3). A group A is called p-semi-simple if \(R_ p\) is a semi-simple ring.
The main result of this paper is Theorem 2.4. 1. Let A be a reduced torsion free group of finite rank and for any prime p A be p-semi-simple. Then R is a hereditary ring and \(A=\sum^{n}_{i=1}\oplus A_ i\) where all groups \(A_ i\) are fully invariant in A, and for all \(i\in \{1,2,...,n\}\) \(E(A_ i)\) are prime rings and \(A_ i\) are p-semi-simple groups for all primes p. 2. If A is p-semi-simple for all primes p and R is a prime ring, then R is a Dedekind domain. 3. A group A with the prime ring R is p-semi-simple for all p if and only if \(A\cong {\mathcal F}\otimes_ cB\) where \({\mathcal F}\) is a finitely generated projective module over an integral domain such that all rings C/pC are completely reducible, B is a torsion free group such that E(B)\(\cong C\).
Reviewer: A.M.Sebel’din

MSC:

20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
20K15 Torsion-free groups, finite rank
16S50 Endomorphism rings; matrix rings
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