Completely decomposable homogeneous quotient divisible abelian groups

E. V. Gordeeva ; A. A. Fomin

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 377-388.

Résumé

L.S. Pontryagin [1], A.G. Kurosh [2], A.I. Mal'cev [3], D. Derry [4], R. Baer [5], R. Beaumont and R. Pierce [6,7] began research on abelian torsion free groups of finite rank. In particular, R. Beaumont and R. Pierce [6] introduced the notion of the quotient divisible torsion free group. The notion of quotient divisible group was extended to the case of mixed groups in the work [8]. It was also proved in [8] that the category of quotient divisible mixed groups with quasi-homomorphisms was dual to the category of torsion-free finite-rank groups with quasi-homomorphisms. A modern version of the duality [8] was obtained in [9, 10]. The categories of groups with quasi-homomorphisms were replaced by the categories of groups with marked bases and with usual homomorphisms such that their matrices with respect to the marked bases consisted of integers. The duality [8] was also extended by S. Breaz and P. Schultz [11] on the class of self-small groups. The mixed quotient divisible groups as well as the self-small groups are in the focus of attention now [12-35].In the present paper we prove two theorems about homogeneous completely decomposable quotient divisible mixed groups. In the first theorem we show that for every basis of such group there exists a decomposition of this group into a direct sum of rank-1 groups such that the elements of the basis are the bases of the corresponding rank-1 groups. Moreover, for every two bases such decompositions are isomorphic. In the second theorem we show that every exact sequence of quotient divisible groups $0\rightarrow B\rightarrow A\rightarrow C\rightarrow 0$ is splitting, if the group $A$ is homogeneous completely decomposable. This theorem is a dual version of the following classic result by R. Baer. Every pure subgroup of a homogeneous completely decomposable torsion free group of finite rank is a direct summand of this group.

Détail

Keywords: abelian groups, direct decompositions, dual categories.

Bibliographie

[1] Pontryagin L. S., “The theory of topological commutative groups”, Ann. of Math., 13:1 (1934), 361–388 | DOI | MR | MR

[2] A. G. Kurosh, “Primitive torsionsfreie abelsche Gruppen von endlichen Range”, Ann. of Math., 38:1 (1937), 175–203 | DOI | MR | Zbl

[3] Malcev A. I., “Torsionsfreie Abelsche Gruppen vom endlichen Rang”, Rec. Math. N.S., 4:1 (1938), 45–68 | Zbl

[4] D. Derry, “Über eine Klasse von abelschen Gruppen”, Proc. London Math. Soc., 1938:43, 490–506 | MR

[5] R. Baer, “Abelian groups without elements of finite order”, Duke Math., 3:1 (1937), 68–122 | DOI | MR

[6] R. A. Beaumont, R. S. Pierce, “Torsion free rings”, Ill. J. Math., 5:1 (1961), 61–98 | DOI | MR | Zbl

[7] R. A. Beaumont, R. S. Pierce, “Torsion free groups of rank two”, Mem. Amer. Math. Soc., 38:1 (1961), 1–41 | MR

[8] A. A. Fomin, W. J. Wickless, “Quotient divisible abelian groups”, Proc. A.M.S., 126:1 (1998), 45–52 | DOI | MR | Zbl

[9] A. A. Fomin, “Invariants for Abelian groups and dual exact sequences”, J. Algebra, 322:7 (2009), 2544–2565 | DOI | MR | Zbl

[10] Yakovlev A. B., “Duality of the categories of torsion-free Abelian groups of finite rank and quotient divisible Abelian groups”, J. Math. Sci., 171:3 (2010), 416–420 | DOI | MR | Zbl

[11] S. Breaz, P. Schultz, “Dualities for self-small groups”, Proc. A.M.S., 140:1 (2012), 69–82 | DOI | MR | Zbl

[12] Davydova O. I., “Rank-1 quotient divisible groups”, J. Math. Sci., 154:3 (2008), 295–300 | DOI | MR | Zbl

[13] Davydova O. I., “Homomorphisms of Rank-1 Quotient Divisible Groups”, J. Math. Sci., 230:3 (2018), 389–391 | DOI | MR | Zbl

[14] Fomin A. A., “To Quotient Divisible Group Theory. I”, J. Math. Sci., 197:5 (2014), 688–697 | DOI | MR | Zbl

[15] Fomin A. A., “On the Quotient Divisible Group Theory. II”, J. Math. Sci., 230:3 (2018), 457–483 | DOI | MR | Zbl

[16] W. J. Wickless, “Direct sums of quotient divisible groups”, Communications in Algebra, 31:1 (2003), 79–96 | DOI | MR | Zbl

[17] U. Albrecht, S. Breaz, C. Vinsonhaler, W. Wickless, “Cancellation properties for quotient divisible groups”, Journal of Algebra, 317:1 (2007), 424–434 | DOI | MR | Zbl

[18] Tsarev A. V., “Modules over the ring of pseudorational numbers and quotient divisible groups”, St. Petersburg Math. J., 18:4 (2007), 657–669 | DOI | MR | Zbl

[19] Lyubimtsev O. V., “Completely decomposable quotient divisible abelian groups with UA-rings of endomorphisms”, Mathematical Notes, 98:1 (2015), 130–137 | DOI | DOI | MR | Zbl

[20] W. Wickless, “Multi-isomorphism for quotient divisible groups”, Houston J. Math., 31:1 (2006), 1–19 | MR

[21] Tsarev A. V., “The module of pseudo-rational relations of a quotient divisible group”, St. Petersburg Math. J., 22:1 (2010), 163–174 | DOI | MR

[22] Tsarev A. V., “Pseudorational rank of a quotient divisible group”, J. Math. Sci., 144:2 (2007), 4013–4022 | DOI | MR | Zbl

[23] Tsarev A. V., “$T$-rings and quotient divisible groups of rank 1”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 24:4 (2013), 50–53

[24] Kryuchkov N. I., “Compact Groups That Are Duals of Quotient Divisible Abelian Groups”, J. Math. Sci., 230:3 (2018), 428–432 | DOI | MR | Zbl

[25] Lyubimtsev O. V., “Characterization of completely decomposable quotient divisible abelian groups by their endomorphism semigroups”, Russian Math. (Iz. VUZ), 61:10 (2017), 65–71 | DOI | MR | MR | Zbl

[26] S. Files, W. Wickless, “Direct Sums of Self-Small Mixed Groups”, J. Algebra, 222:1 (1999), 1–16 | DOI | MR | Zbl

[27] U. Albrecht, B. Wickless, “Finitely generated and cogenerated $QD$ groups”, Rings, modules, algebras, and abelian, 236:1 (2004), 13–26 | MR | Zbl

[28] U. Albrecht, S. Breaz, W. Wickless, “Self-small abelian groups”, Bulletin of the Australian Mathematical Society, 80:2 (2009), 205–216 | DOI | MR | Zbl

[29] A. A. Fomin, W. Wickless, “Self-small mixed abelian groups $G$ with $G/T(G)$ finite rank divisible”, Communications in Algebra, 26:11 (1998), 3563–3580 | DOI | MR | Zbl

[30] U. Albrecht, S. Breaz, W. Wickless, “Purity and Self-Small Groups”, Communications in Algebra, 35:11 (2007), 3789–3807 | DOI | MR | Zbl

[31] J. Zemlicka, “When Products of Self-Small Modules are Self-Small”, Communications in Algebra, 36:7 (2008), 2570–2576 | DOI | MR | Zbl

[32] S. Breaz, J. Zemlicka, “When every self-small module is finitely generated”, J. Algebra, 315:2 (2007), 885–893 | DOI | MR | Zbl

[33] U. Albrecht, S. Breaz, “A note on self-small modules over $RM$-domains”, J. Algebra and Its Applications, 13:1 (2014), 8 pp. | DOI | MR

[34] A. A. Fomin, “Quotient divisible and almost completely decomposable groups”, Models, Modules and Abelian Groups, in Memory of A. L. S. Corner, de Gruyter, Berlin–New York, 2008, 147–167 | MR | Zbl

[35] Fomin A. A., “Almost completely decomposable groups”, Proceedings of the XIII International conference devoted to the 85-th Birthday of Professor S.S. Ryshkov (May 25–30, 2015, Tula), 2015, 49–52 (Russian)

[36] L. Fuks, Beskonechnye abelevy gruppy, v. 1, Mir, M., 1974; v. 2, 1977; L. Fuchs, Infinite Abelian Groups, v. 1, Academic Press, New York, 1970 ; v. 2, 1973 | MR | Zbl

[37] A. Corner, “A note on rank and direct decomposition of torsion-free abelian groups”, Proc. Cambridge Philos. Soc., 66:1 (1969), 239–240 | DOI | MR | Zbl