Co-Hopfian Abelian groups

E. V. Kaigorodov   ; S. M. Chedushev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 4 (2015), pp. 21-33

Résumé

In recent years, the interest in co-Hopfian algebraic systems has been growing steadily, with a great number of publications on the topic. However, the studies on co-Hopfian Abelian groups are represented only by individual works. It is therefore natural that there is quite a lot of interesting and important but still open questions related to co-Hopfian Abelian groups. One of these concerns the description of co-Hopfian groups in specific classes of Abelian groups. Consequently, the study of co-Hopfian Abelian groups and their properties is of particular interest. The first section of this paper contains a detailed review of known results on co-Hopfian algebraic systems, the primary emphasis being on co-Hopfian Abelian groups. Special attention is paid to co-Hopfian rings and modules. Some of the major results obtained by specialists in the last half-century are considered in detail. In the second section we obtain the general properties of co-Hopfian Abelian groups. For instance, we prove the co-Hopficity of direct summands of a co-Hopfian Abelian group. We point to one of the cases in which the co-Hopficity of an Abelian group should follow from the co-Hopficity of direct summands in the decomposition of this group. Finally, we give a necessary and sufficient condition of the co-Hopficity of a direct sum of an arbitrary number of Abelian groups on one assumption.

Détail

Keywords: Abelian group, co-Hopfian group, direct sum, fully invariant subgroup, generalized matrix ring.

Bibliographie

[1] Baer R., “Groups without proper isomorphic quotient groups”, Bull. Amer. Math. Soc., 50:4 (1944), 267–278 | DOI | MR | Zbl

[2] Gonzales-Acuna F., Whitten W., “Embeddings of three-manifold groups”, Mem. Amer. Math. Soc., 99, no. 474, 1992

[3] Potyagailo L., Wang S., “3-manifolds with co-Hopfian fundamental group”, St. Petersburg. Math. J., 11:5 (2000), 861–881 | MR | Zbl

[4] Ohshika K., Potyagailo L., “Self-embeddings of Kleinian groups”, Ann. Sci. Ecole Norm. Sup., 31:3 (1998), 329–343 | MR | Zbl

[5] Sela Z., “Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie algebras, II”, Geometric and Fundamental Analysis, 7:3 (1997), 561–593 | DOI | MR | Zbl

[6] Wang S. C., Wu Y. Q., “Covering invariant and cohopficity of 3-manifold groups”, Proc. London Math. Soc., 68 (1994), 203–224 | DOI | MR | Zbl

[7] Wang S. C., Zhou Q., “Embeddings of Kleinian groups with torsion”, Acta Math. Sin. (Engl. Ser.), 17:1 (2001), 21–34 | DOI | MR | Zbl

[8] Deo S., Varadarajan K., “Hopfian and co-Hopfian groups”, Bull. Austral. Math. Soc., 56:1 (1997), 17–24 | DOI | MR | Zbl

[9] Bell R. W., Margalit D., “Braid groups and the co-Hopfian property”, J. Algebra, 303:1 (2006), 275–294 | DOI | MR | Zbl

[10] Li Y., “On the Cohopficity of the Direct Product of Cohopfian Groups”, Comm. Algebra, 35:10 (2007), 3226–3235 | DOI | MR | Zbl

[11] Endimioni G., “Hopficity and co-Hopficity in soluble groups”, Ukr. Math. J., 56:10 (2004), 1594–1601 | DOI | MR | Zbl

[12] Cain A. J., Maltcev V., “Hopfian and co-hopfian subsemigroups and extensions”, Demonstratio Mathematica, 47:4 (2014), 791–804 | DOI | MR | Zbl

[13] Varadarajan K., “Hopfian and co-Hopfian objects”, Publ. Math., 36 (1992), 293–317 | DOI | MR | Zbl

[14] Varadarajan K., “Some recent results on Hopficity, co-Hopficity and related properties”, International Symposium on Ring Theory, Trends in Math., Birkhauser–Boston, 2002, 371–392 | MR

[15] Varadarajan K., “Anti Hopfian and anti co-Hopfian modules”, Contemporary Mathemathics, 456, 2008, 205–218 | DOI | MR | Zbl

[16] Xue W., “Hopfian modules and co-Hopfian modules”, Comm. Algebra, 23:4 (1995), 1219–1229 | DOI | MR | Zbl

[17] Asgari Sh., “On weakly co-Hopfian modules”, B. Iran. Math. Soc., 33:1 (2007), 65–72 | MR | Zbl

[18] Haghany A., “Hopficity and co-Hopficity for Morita contexts”, Comm. Algebra, 27 (1999), 477–492 | DOI | MR | Zbl

[19] Haghany A., Vedadi M. R., “Modules whose injective endomorphisms are essential”, J. Algebra, 243 (2001), 765–779 | DOI | MR | Zbl

[20] Asgari Sh., Haghany A., Vedadi M. R., “Quasi co-Hopfian modules and applications”, Comm. Algebra, 36:5 (2008), 1801–1816 | DOI | MR | Zbl

[21] Asgari Sh., Haghany A., “Densely co-Hopfian modules”, J. Algebra Appl., 9:6 (2010), 989–1000 | DOI | MR | Zbl

[22] Fan Y., Liu Z., “Co-Hopfian modules of generalized inverse polynomials”, Acta Math. Sin. (Engl. Ser.), 17:3 (2001), 431–436 | DOI | MR | Zbl

[23] Gang Y., Liu Z., “On Hopfian and co-Hopfian modules”, Vietnam J. Math., 35:1 (2007), 73–80 | MR | Zbl

[24] Gang Y., Liu Z., “On generalizations of Fitting modules”, Indian J. Math., 51:1 (2009), 85–99 | MR | Zbl

[25] Gang Y., Liu Z., “Notes on generalized Hopfian and weakly co-Hopfian modules”, Comm. Algebra, 38 (2010), 3556–3566 | DOI | MR | Zbl

[26] Ghorbani A., Haghany A., “Generalized Hopfian modules”, J. Algebra, 255 (2002), 324–341 | DOI | MR | Zbl

[27] Ghorbani A., Haghany A., “Duality for weakly co-Hopfian and generalized Hopfian modules”, Comm. Algebra, 31 (2003), 2811–2817 | DOI | MR | Zbl

[28] Wang Y., “Generalizations of Hopfian and co-Hopfian modules”, Int. J. Math. Sci., 9 (2005), 1455–1460 | DOI | MR | Zbl

[29] Divaani-Aazar K., Mafi A., “Hopfian and co-Hopfian modules over commutative rings”, Vietnam J. Math., 35:3 (2007), 275–283 | MR | Zbl

[30] Hmaimou A., Kaidi A., Campos E. S., “Generalized Fitting modules and rings”, J. Algebra, 308 (2007), 199–214 | DOI | MR | Zbl

[31] Aydoğdu P., Özcan A. C., “Semi co-Hopfian and semi Hopfian modules”, East-West J. Math., 10:1 (2008), 57–72 | MR

[32] Yan X. F., Liu Z., “Extensions of generalized Fitting modules”, J. Math. Res. Exp., 30:3 (2010), 407–414 | MR | Zbl

[33] Jiao Y. J., “Semi Hopfian and semi co-Hopfan modules over generalized power series rings”, Int. J. Algebra, 6:5–8 (2012), 209–218 | MR | Zbl

[34] Wang X., Li T., “A Generalization of Weakly Co-hopfian Modules”, Int. Math. Forum, 9:6 (2014), 255–258 | MR

[35] Diallo E., Maaouia M., Sanghare M., “Hopfian Objects, Cohopfian Objects in the Category of Complexes of Left A-Modules”, Int. Math. Forum, 8:39 (2013), 1903–1920 | MR | Zbl

[36] Diallo E., Maaouia M., Sanghare M., “Strongly Hopfian and Strongly Cohopfian Objects in the Category of Complexes of Left A-Modules”, J. Math. Res., 6:3 (2014), 81–90 | DOI

[37] Irwin J. M., Takashi J., “A quasi-decomposable Abelian group without proper isomorphic quotient groups and proper isomorphic subgroups”, Pacif. J. Math., 29:1 (1969), 151–160 | DOI | MR | Zbl

[38] Takashi J., Irwin J. M., “A quasi-decomposable Abelian group without proper isomorphic quotient groups and proper isomorphic subgroups, 2”, J. Fac. Sci. Hokkaido Univ., 20:4 (1969), 194–203 | MR

[39] Goldsmith B., Gong K., “On super and hereditarily Hopfian and co-Hopfian Abelian groups”, Arch. Math., 99:1 (2012), 1–8 | DOI | MR | Zbl

[40] Goldsmith B., Gong K., “On adjoint entropy of Abelian groups”, Comm. Algebra, 40 (2012), 972–987 | DOI | MR | Zbl

[41] Goldsmith B., Gong K., A note on Hopfian and co-Hopfian Abelian groups, AMS forthcoming, Dublin, 2012, 9 pp. | MR

[42] Goldsmith B., Gong K., “On some generalizations of Hopfian and co-Hopfian Abelian groups”, Acta Math. Hung., 139:4 (2013), 393–398 | DOI | MR | Zbl

[43] Dikranjan D., Goldsmith B., Salce L., Zanardo P., “Algebraic entropy for Abelian groups”, Trans. Amer. Math. Soc., 361:7 (2009), 3401–3434 | DOI | MR | Zbl

[44] Beaumont R. A., “Groups with isomorphic proper subgroups”, Bull. Amer. Math. Soc., 51 (1945), 381–387 | DOI | MR | Zbl

[45] Beaumont R. A., Pierce R. S., “Partly transitive modules and modules with proper isomorphic submodules”, Trans. Amer. Math. Soc., 91 (1959), 209–219 | DOI | MR | Zbl

[46] Beaumont R. A., Pierce R. S., “Isomorphic direct summands of Abelian groups”, Math. Annal., 153 (1964), 21–37 | DOI | MR

[47] Grinshpon S. Ya., Nikol'skaya (Savinkova) M. M., “IF-gruppy”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2010, no. 1(9), 5–14 (in Russian)

[48] Grinshpon S. Ya., Nikol'skaya (Savinkova) M. M., “Primarnye IF-gruppy”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2011, no. 3(15), 25–31 (in Russian) | MR

[49] Grinshpon S. Ya., Nikol'skaya (Savinkova) M. M., “Sobstvennye vpolne kharakteristicheskie podgruppy grupp bez krucheniya, izomorfnye samoy gruppe”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2012, no. 1(17), 25–30 (in Russian) | MR

[50] Grinshpon S. Ya., Nikol'skaya (Savinkova) M. M., “Periodicheskie IF-gruppy”, Fundament. i prikl. matem., 17:8 (2012), 47–58 (in Russian)

[51] Kaygorodov E. V., “Khopfovy abelevy gruppy”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2012, no. 2(18), 5–12 (in Russian)

[52] Kaygorodov E. V., “O dvukh klassakh khopfovykh abelevykh grupp”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2013, no. 2(22), 22–33 (in Russian)

[53] Kaygorodov E. V., “Khopfovy vpolne razlozhimye gruppy bez krucheniya”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2013, no. 4(24), 24–28 (in Russian)

[54] Kaigorodov E. V., “Hopfian algebraically compact Abelian groups”, Algebra and Logic, 52:6 (2014), 442–447 | DOI | MR | Zbl

[55] Crawley P., “An infinite primary Abelian group without proper isomorphic subgroups”, Bull. Amer. Math. Soc., 68 (1962), 463–467 | DOI | MR | Zbl

[56] Hill P., Megibben Ch., “On primary groups with countable basic subgroups”, Trans. Amer. Math. Soc., 124:1 (1966), 49–59 | DOI | MR | Zbl

[57] Monk G. S., “Abelian $p$-groups without proper isomorphic pure dense subgroups”, Ill. J. Math., 14:1 (1970), 164–177 | MR | Zbl

[58] Goldsmith B., Ohogain S., Wallutis S., “Quasi-minimal groups”, Proc. Amer. Math. Soc., 132:8 (2004), 2185–2195 | DOI | MR | Zbl

[59] Chekhlov A. R., Danchev P. V., “On Abelian Groups having all proper fully invariant subgroups isomorphic”, Comm. Algebra, 2015 (to appear) | MR

[60] Kurosh A. G., The Theory of Groups, 2 vols., Chelsea Publishing Company, New York, 1960, 612 pp. | MR | MR

[61] Gratzer G., General Lattice Theory, Academic Press, New York, 1978, 404 pp. | MR | MR | Zbl

[62] Krylov P. A., Mikhalev A. V., Tuganbaev A. A., Endomorphism Rings of Abelian Groups, Kluwer Academic Publishers, Dordrecht, 2003, 442 pp. | MR | Zbl

[63] Fuks L., Beskonechnye abelevy gruppy, v. 1, Mir Publ., M., 1974, 336 pp. (in Russian)