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Pure subgroups of \(A\)-projective groups. (English) Zbl 0814.20038

The following example will illustrate the topic considered by the authors. Let \(A =\bigoplus \mathbb{Z}_ p\) be the direct sum of the abelian groups \(\mathbb{Z}_ p\), the integers localized at the prime \(p\), where \(p\) runs over all primes. The endomorphism ring \(E(A) \cong \prod_ p \mathbb{Z}_ p\) of \(A\) is strongly non-singular and semi-hereditary. Recall that a ring \(E(A)\) is strongly non-singular if all finitely generated non-singular (right) \(E(A)\)-modules are the finitely generated submodules of free modules. Moreover \(A\) is flat over \(E(A)\) and satisfies the following two conditions. “\(A\) splits pure \(A\)-socles”, i.e. all pure \(A\)-generated subgroups \(U\) (\(U = \sum\{\varphi(A): \varphi \in \text{Hom}(A,U)\})\) of \(A\)-projective groups \(D\) of finite rank (\(D\) is a summand of \(\bigoplus_ n A\)) are summands. “\(A\) is not quasi-splitting \(A\)-socles”, i.e. there are subgroups \(U\) as above (but not pure) in \(D\) which are not even quasi-summands of \(D\).
The paper gives a detailed analysis of the indicated properties for general torsion-free \(A\) and provides characterizations of them partly in terms of \(E(A)\)-modules (Theorem 2.1) or ring theoretically. Here is one of the results which can be applied to derive the properties of the above example. Corollary 2.3: If \(A\) is torsion-free abelian and faithful as \(E(A)\)-module, then the following are equivalent: (a) \(A\) splits pure \(A\)-socles and is a flat \(E(A)\)-module. (b) \(E(A)\) is (right) semi- hereditary and the quasi-endomorphism ring \(\mathbb{Q} E(A)\) is semi-simple artinian. A special case is Baer’s splitting theorem for pure subgroups of homogeneous completely decomposable groups of finite rank.
Reviewer: R.Göbel (Essen)

MSC:

20K25 Direct sums, direct products, etc. for abelian groups
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
16S50 Endomorphism rings; matrix rings
20K27 Subgroups of abelian groups
16D40 Free, projective, and flat modules and ideals in associative algebras
20K20 Torsion-free groups, infinite rank
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[1] U. Albrecht, Endomorphism rings and a generalization of torsion-freeness and purity,Comm. in Algebra,17 (1989), 1101–1135. · Zbl 0691.20040 · doi:10.1080/00927878908823776
[2] U. Albrecht, Abelian groups which are faithfully flat as modules over their endomorphism rings,Res. der Mathematik,17 (1990), 179–201. · Zbl 0709.20031
[3] U. Albrecht, On the quasi-splitting of exact sequences,J. of Algebra,144 (1991), 344–358. · Zbl 0749.20029 · doi:10.1016/0021-8693(91)90108-K
[4] U. Albrecht,Abelian groups with semi-simple Artinian quasi-endomorphism ring; to appear inRocky Mount. J. of Math. (1994).
[5] U. Albrecht and H. P. Goeters, A dual to Baer’s Lemma,Proc. Amer. Math. Soc.,105 (1989), 817–826. · Zbl 0681.20036
[6] D. M. Arnold,Finite Rank Torsion-Free Abelian Groups and Rings; Springer Lecture Notes in Mathematics 931; Springer Verlag, (Berlin, Heidelberg, New York, 1982). · Zbl 0493.20034
[7] D. M. Arnold,Abelian groups flat over their endomorphism ring; preprint.
[8] D. M. Arnold and L. Lady, Endomorphism rings and direct sums of torsion-free abelian groups,Trans. Amer. Math.,11 (1975), 225–237. · Zbl 0329.20033 · doi:10.1090/S0002-9947-1975-0417314-1
[9] R. Baer, Abelian groups without elements of finite order,Duke Math. J.,3 (1937), 68–122. · Zbl 0016.20303 · doi:10.1215/S0012-7094-37-00308-9
[10] M. Dugas and R. Göbel, Every cotorsion-free ring is an endomorphism ring,Proc. London Math. Soc.,45 (1982), 319–336. · Zbl 0506.16022 · doi:10.1112/plms/s3-45.2.319
[11] T. Faticoni and H. P. Goeters, On torsion-free Ext,Comm. in Alg.,16 (1988), 1853–1876. · Zbl 0667.20042 · doi:10.1080/00927878808823664
[12] M. Huber and R. B. Warfield, Homomorphisms between Cartesian powers of an abelian group, inAbelian Groups, Proceedings Oberwolfach 1981; Springer Lecture Notes in Mathematics 874; Springer Verlag (Berlin, Heidelberg, New York, 1981), pp. 202–227. · Zbl 0484.20024
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