Let R be a commutative domain. We prove that an R-module B is projective if and only if Ext 1 R (B,T) = 0 for any torsion module T. This answers in the affirmative a question raised by Kaplansky in 1962.

/pdf/a-solution-to-the-baer-splitting-problem-kzyk4lwxtc.pdf

Infinite Abelian groups

A solution to the Baer splitting problem

We discuss faithfully flat abelian groups in conjunction with the following: A class C of abelian groups with full A-socle is A-balanced closed if it is closed with respect to finite direct sums and subgroups with full A-socle, ker ϕ e C for all ϕ e Horn (G, H) and G, H e C, and A is projective with respect to all exact sequences of elements of C. A self-small group A admits an A-balanced closed class C which contains ⊕IA for all index-sets I exactly if it is faithfully flat as an E(A)-module. We show that Corner's as well as Dugas' and Gobel's realization theorems yield abelian groups that are faithfully flat as E(A)-modules. Several applications of these results are given, some of which yield an answer to part a of Fuchs' Problem 84 and a partial respond to part c of the same problem.

Endomorphism Rings of Faithfully Flat Abelian Groups

Each valuated module B with B(5) = 0 is a direct sum of simply presented valuated modules and copies of two valuated modules which come from (finite) hung trees. There are infinite-rank indecomposable valuated modules B with B(6) = 0.

/pdf/subgroups-of-p5-bounded-groups-3udhkbyyf7.pdf

Subgroups of p5-bounded groups

https://msp.org/pjm/1974/53-1/pjm-v53-n1-p10-s.pdf

Some aspects of $T$-nilpotence.

© Andrée C. Ehresmann et les auteurs, 1986, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Infinite Abelian groups

Citations

A solution to the Baer splitting problem

Endomorphism Rings of Faithfully Flat Abelian Groups

Subgroups of p5-bounded groups

Some aspects of $T$-nilpotence.

Closure operators, monomorphisms and epimorphisms in categories of groups

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