The Lattice of Fully Invariant Subgroups of the Cotorsion Hull

ABSTRACT

The paper considers the lattice of fully invariant subgroups of the cotorsion hull when a separable primary group T is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation difined on the set of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice. The use of the relation essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group is isomorphic to the lattice of filters of the lower semilattice.

KEYWORDS

Lattice of Fully Invariant Subgroups; Direct Sum of Torsion-Complete Groups; Cotorsion Hull

Lattice of Fully Invariant Subgroups; Direct Sum of Torsion-Complete Groups; Cotorsion Hull

Cite this paper

T. Kemoklidze, "The Lattice of Fully Invariant Subgroups of the Cotorsion Hull,"*Advances in Pure Mathematics*, Vol. 3 No. 8, 2013, pp. 670-679. doi: 10.4236/apm.2013.38090.

T. Kemoklidze, "The Lattice of Fully Invariant Subgroups of the Cotorsion Hull,"

References

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http://dx.doi.org/10.1112/plms/s2-39.1.481

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http://dx.doi.org/10.1307/mmj/1029001528

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http://dx.doi.org/10.1007/BF01214169

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http://dx.doi.org/10.1007/s10958-005-0245-5

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[13] A. I. Moskalenko, “Cotorsion Hull of a Separable Group,” Algebra i Logika, Vol. 28, No. 2, 1989, pp. 207-226 (in Russian); Translation in Algebra and Logic, Vol. 28, No. 2, 1989, pp. 139-151 (1990).

[14] D. K. Harrison, “Infinite Abelian Groups and Homological Methods,” Annals of Mathematics, Vol. 69, No. 2, 1959, pp. 366-391. http://dx.doi.org/10.2307/1970188

[15] S. Bazzoni and L. Salce, “An Independence Result on Cotorsion Theories over Valuation Domains,” Journal of Algebra, Vol. 243, No. 1, 2001, pp. 294-320.

http://dx.doi.org/10.1006/jabr.2001.8800

[16] S. Bazzoni and J. Stóvicek, “Sigma-Cotorsion Modules over Valuation Domains,” Forum Mathematicum, Vol. 21, No. 5, 2009, pp. 893-920.

http://dx.doi.org/10.1515/FORUM.2009.044

[17] R. Gobel, S. Shelah and S. L. Wallutis, “On the Lattice of Cotorsion Theories,” Journal of Algebra, Vol. 238, No, 1, 2001, pp. 292-313.

http://dx.doi.org/10.1006/jabr.2000.8619

[18] M. Hovey, “Cotorsion Pairs, Model Category Structures, and Representation Theory,” Mathematische Zeitschrift, Vol. 241, No. 3, 2002, pp. 553-592.

http://dx.doi.org/10.1007/s00209-002-0431-9

[19] W. May and E. Toubassi, “Endomorphisms of Abelian Groups and the Theorem of Baer and Kaplansky,” Journal of Algebra, Vol. 43, No. 1, 1976, pp. 1-13.

http://dx.doi.org/10.1016/0021-8693(76)90139-3

[20] T. Kemoklidze, “On the Full Transitivity of a Cotorsion Hull,” Georgian Mathematical Journal, Vol. 13, No. 1, 2006, pp.79-84.

[21] T. Kemoklidze, “The Lattice of Fully Invariant Subgroups of a Cotorsion Hull,” Georgian Mathematical Journal, Vol. 16, No. 1, 2009, pp. 89-104.

[22] T. Kemoklidze, “On the Full Transitivity and Fully Invariant Subgroups of Cotorsion Hulls of Separable pGroups,” Journal of Mathematical Sciences (New York), Vol. 155, No. 5, 2008, pp. 748-786.

http://dx.doi.org/10.1007/s10958-008-9240-y

[1] L. Fuchs, “Infinite Abelian Groups. I,” Academic Press, New York, London, 1970.

[2] L. Fuchs, “Infinite Abelian Groups. II,” Academic Press, New York, London, 1973.

[3] R. Baer, “Type of Elements and Characteristic Subgroups of Abelian Groups,” Proceedings London Mathematical Society, Vol. s2-39, No. 1, 1935, pp. 481-514.

http://dx.doi.org/10.1112/plms/s2-39.1.481

[4] I. Kaplansky, “Infinite Abelian Groups,” The University of Michigan Press, Ann Arbor, 1969.

[5] R. S. Linton, “On Fully Invariant Subgroups of Primary Abelian Groups,” Michigan Mathematical Journal, Vol. 22, No. 3, 1976, pp. 281-284.

http://dx.doi.org/10.1307/mmj/1029001528

[6] J. D. Moore and E. J. Hewett, “On Fully Invariant Subgroups of Abelian p-Groups,” Commentarii Mathematici Universitatis Sancti Pauli, Vol. 20, 1971-1972, pp. 97-106.

[7] R. S. Pierce, “Homomorphisms of Primary Abelian Groups,” In: Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman and Co., Chicago, 1963, pp. 215-310.

[8] A. R. Chekhlov, “On Projective Invariant Subgroups of Abelian Groups,” Vestnik Tomskogo Gosudarstvennogo Universiteta Matematika i Mekhanika, Vol. 2009, No. 1, 2009, pp. 31-36 (in Russian).

[9] R. Gobel, “The Characteristic Subgroups of the Baer-Specker Group,” Mathematische Zeitschrift, Vol. 140, No. 3, 1974, pp. 289-292.

http://dx.doi.org/10.1007/BF01214169

[10] S. Ya. Grinshpon and P. A. Krylov, “Fully invariant Subgroups, Full Transitivity, and Homomorphism Groups of Abelian Groups. Algebra,” Journal of Mathematical Sciences (New York), Vol. 128, No. 3, 2005, pp. 2894-2997.

http://dx.doi.org/10.1007/s10958-005-0245-5

[11] A. Mader, “The Fully Invariant Subgroups of Reduced Algebraically Compact Groups,” Publicationes Mathematicae Debrecen, Vol. 17, 1970, pp. 299-306.

[12] V. M. Misyakov, “On Full Transitivity of Reduced Abelian Groups,” In: Abelian Groups and Modules, No. 11, 12 (Russian), Tomsk State University, Tomsk, 1994, pp. 134-156 (in Russian).

[13] A. I. Moskalenko, “Cotorsion Hull of a Separable Group,” Algebra i Logika, Vol. 28, No. 2, 1989, pp. 207-226 (in Russian); Translation in Algebra and Logic, Vol. 28, No. 2, 1989, pp. 139-151 (1990).

[14] D. K. Harrison, “Infinite Abelian Groups and Homological Methods,” Annals of Mathematics, Vol. 69, No. 2, 1959, pp. 366-391. http://dx.doi.org/10.2307/1970188

[15] S. Bazzoni and L. Salce, “An Independence Result on Cotorsion Theories over Valuation Domains,” Journal of Algebra, Vol. 243, No. 1, 2001, pp. 294-320.

http://dx.doi.org/10.1006/jabr.2001.8800

[16] S. Bazzoni and J. Stóvicek, “Sigma-Cotorsion Modules over Valuation Domains,” Forum Mathematicum, Vol. 21, No. 5, 2009, pp. 893-920.

http://dx.doi.org/10.1515/FORUM.2009.044

[17] R. Gobel, S. Shelah and S. L. Wallutis, “On the Lattice of Cotorsion Theories,” Journal of Algebra, Vol. 238, No, 1, 2001, pp. 292-313.

http://dx.doi.org/10.1006/jabr.2000.8619

[18] M. Hovey, “Cotorsion Pairs, Model Category Structures, and Representation Theory,” Mathematische Zeitschrift, Vol. 241, No. 3, 2002, pp. 553-592.

http://dx.doi.org/10.1007/s00209-002-0431-9

[19] W. May and E. Toubassi, “Endomorphisms of Abelian Groups and the Theorem of Baer and Kaplansky,” Journal of Algebra, Vol. 43, No. 1, 1976, pp. 1-13.

http://dx.doi.org/10.1016/0021-8693(76)90139-3

[20] T. Kemoklidze, “On the Full Transitivity of a Cotorsion Hull,” Georgian Mathematical Journal, Vol. 13, No. 1, 2006, pp.79-84.

[21] T. Kemoklidze, “The Lattice of Fully Invariant Subgroups of a Cotorsion Hull,” Georgian Mathematical Journal, Vol. 16, No. 1, 2009, pp. 89-104.

[22] T. Kemoklidze, “On the Full Transitivity and Fully Invariant Subgroups of Cotorsion Hulls of Separable pGroups,” Journal of Mathematical Sciences (New York), Vol. 155, No. 5, 2008, pp. 748-786.

http://dx.doi.org/10.1007/s10958-008-9240-y