Skip to main content
Log in

Hereditary Endomorphism Rings of Mixed Abelian Groups

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

Let A be a finite-rank, torsion-free, self-small mixed abelian sp-group and let E(A) be the endomorphism ring of A. We give conditions for right and left heredity of E(A). A ring is right hereditary if each of its right ideals is projective. We also find the structure of one-sided ideals of E(A).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Markov V. T., Mikhalëv A. V., Skornyakov L. A., and Tuganbaev A. A., “Endomorphism rings of modules and structures of submodules,” in: Algebra. Topology. Geometry [in Russian], VINITI, Moscow, 1983, 21, pp. 183–254. (Itogi Nauki i Tekhniki.)

    Google Scholar 

  2. Fuchs L., Infinite Abelian Groups. Vol. 2 [Russian translation], Mir, Moscow (1977).

    Google Scholar 

  3. Arnold D. M. and Lady E. L., “Endomorphism rings and direct sums of torsion-free abelian groups,” Trans. Amer. Math. Soc., 211, 225–237 (1975).

    Google Scholar 

  4. Huber M. and Warfield R. B. Jr., “Homomorphisms between cartesian powers of an abelian group,” Lecture Notes Math., 874, 202–227 (1981).

    Google Scholar 

  5. Albrecht U., “Baer's lemma and Fuchs' problem 84a,” Trans. Amer. Math. Soc., 293, 565–582 (1986).

    Google Scholar 

  6. Faticoni T. G., “On the lattice of right ideals of the endomorphism ring of an abelian group,” Bull. Austral. Math. Soc., 38, No. 2, 273–291 (1988).

    Google Scholar 

  7. Krylov P. A., “On one class of abelian groups with hereditary endomorphism rings,” Sibirsk. Mat. Zh., 28, No. 6, 60–65 (1987).

    Google Scholar 

  8. Krylov P. A., “Torsion-free abelian groups with hereditary endomorphism rings,” Algebra i Logika, 27, No. 3, 295–304 (1988).

    Google Scholar 

  9. Krylov P. A., “On modules with hereditary endomorphism rings,” Uspekhi Mat. Nauk, 45, No. 4, 159–160 (1990).

    Google Scholar 

  10. Glaz S. and Wickless W., “Regular and principal projective endomorphism rings of mixed abelian groups,” Comm. Algebra, 22, No. 4, 1161–1176 (1994).

    Google Scholar 

  11. Albrecht U. F., Goeters H. P., and Wickless W., “The at dimension of mixed abelian groups as E-modules,” Rocky Mountain J. Math., 25, 569–590 (1995).

    Google Scholar 

  12. Albrecht U. F., “Mixed abelian groups with artinian quasi-endomorphism ring,” Comm. Algebra, 25, No. 11, 3497–3511 (1997).

    Google Scholar 

  13. Fomin A. and Wickless W., “Self-small mixed abelian groups G with G/T(G) finite rank divisible,” Comm. Algebra, 26, No. 11, 3563–3580 (1998).

    Google Scholar 

  14. Krylov P. A., “Mixed abelian groups as modules over their endomorphism rings,” Fund. Appl. Math., 6, No. 3, 793–812 (2000).

    Google Scholar 

  15. Krylov P. A., Pakhomova E. G., and Podberezina E. I., “On one class of mixed abelian groups,” Vestnik Tomsk. Univ., 269, 29–34 (2000).

    Google Scholar 

  16. Krylov P. A. and Pakhomova E. G., “Abelian groups and regular modules,” Mat. Zametki, 69, No. 3, 402–411 (2001).

    Google Scholar 

  17. Wickless W., “A functor from mixed group to torsion-free group,” Contemp. Math., 171, 407–419 (1995).

    Google Scholar 

  18. Fomin A. and Wickless W., “Categories of mixed and torsion-free abelian groups,” in: Abelian Groups and Modules, Kluwer, Boston, 1995, pp. 185–192.

    Google Scholar 

  19. Fomin A. and Wickless W., “Quotient divisible abelian groups,” Proc. Amer. Math. Soc., 126, No. 1, 45–52 (1998).

    Google Scholar 

  20. Warfield R. B. Jr., “The structure of mixed abelian groups,” Lecture Notes Math., 616, 1–38 (1977).

    Google Scholar 

  21. Faith C., Algebra: Rings, Modules, and Categories. Vol. 1, 2 [Russian translation], Mir, Moscow (1977, 1979).

    Google Scholar 

  22. Fomin A. A., “Some mixed abelian groups as modules over the ring of pseudo-rational numbers,” Abelian Groups and Modules, Trends in Math., 87–100 (1999).

  23. Pierce R. S., “E-modules,” Contemp. Math., 87, 221–240 (1989).

    Google Scholar 

  24. Krylov P. A. and Prikhodovski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{i} \) M. A., “Generalized T-modules and E-modules,” in: Universal Algebra and Some of Its Applications, Proceedings of the International Seminar Dedicated to the Memory of L. A. Skornyakov, Volgograd, 2000, pp. 153–169.

    Google Scholar 

  25. Arnold D. M. and Murley C. E., “Abelian groups A such that Hom(A, — ) preserves direct sums of copies of A,” Pacific J. Math., 56, No. 1, 7–20 (1975).

    Google Scholar 

  26. Herstein I. N., Noncommutative Rings [Russian translation], Mir, Moscow (1972).

    Google Scholar 

  27. Puninski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{i} \) G. E. and Tuganbaev A. A., Rings and Modules [in Russian], Soyuz, Moscow (1998).

    Google Scholar 

  28. Albrecht U. F., “Abelian groups at as modules over their endomorphism ring,” Comm. Algebra, 21, 3403–3423 (1993).

    Google Scholar 

  29. Bourbaki N., Commutative Algebra [Russian translation], Mir, Moscow (1971).

    Google Scholar 

  30. Osofsky B. L., “Noninjective cyclic modules,” Proc. Amer. Math. Soc., 19, 1383–1384 (1968).

    Google Scholar 

  31. Drozd Yu. A. and Kirichenko V. V., Finite-Dimensional Algebras [in Russian], Vishcha Shkola, Kiev (1980).

    Google Scholar 

  32. Krylov P. A. and Pakhomova E. G., “Abelian groups as injective modules over endomorphism rings,” Fund. Appl. Math., 4, No. 4, 1365–1384 (1998).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Krylov, P.A. Hereditary Endomorphism Rings of Mixed Abelian Groups. Siberian Mathematical Journal 43, 83–91 (2002). https://doi.org/10.1023/A:1013876621908

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1013876621908

Keywords

Navigation