Skip to main content
SpringerLink
Log in

Ring and Module Hulls

  • Chapter
Extensions of Rings and Modules

  • 1159 Accesses

Abstract

The existence and usefulness of the injective hull of a module is well known. In this chapter several hulls for a ring or a module which are essential extensions that are “minimal”, in some sense, with respect to being contained in some designated class of rings or modules are introduced. The definition of hulls includes most of the known hulls (e.g., injective, quasi-injective, continuous, quasi-continuous, etc.), as well as, some relatively newer ones (e.g., quasi-Baer, right FI-extending, right p.q.-Baer, idempotent closure, right duo). The transfer of information between these hulls and their base rings or modules is discussed.

In Sects. 8.1 and 8.2, basic results and examples are provided. In Sect. 8.3, the maximal right ring of quotients for any ring is shown to enjoy a generalized extending property for a particular set of ideals. A consequence of this result is that every ring has a hull in the idempotent closure class of rings. For a semiprime ring, its idempotent closure hull coincides with the quasi-Baer ring hull and the FI-extending ring hull. In the fourth section, our focus is on modules. An in-depth-treatment is given to the known results on the existence of continuous hulls. Then an FI-extending hull is shown to exist for every finitely generated projective module over a semiprime ring. Finally, in contrast to essential extensions of extending modules, both the extending and the FI-extending properties are shown to transfer from a module to its rational hull.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Akalan, E., Birkenmeier, G.F., Tercan, A.: Goldie extending modules. Commun. Algebra 37, 663–683 (2009). Corrigendum 38, 4747–4748 (2010). Corrigendum 41, 2005 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Akalan, E., Birkenmeier, G.F., Tercan, A.: Goldie extending rings. Commun. Algebra 40, 423–428 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andrunakievic, V.A.: Radicals of associative rings I. Transl. Am. Math. Soc. 52, 95–128 (1966)

    Google Scholar 

  4. Armendariz, E.P., Birkenmeier, G.F., Park, J.K.: Rings containing ideals with bounded index. Commun. Algebra 30, 787–801 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beidar, K., Wisbauer, R.: Strongly and properly semiprime modules and rings. In: Jain, S.K., Rizvi, S.T. (eds.) Ring Theory. Proc. Ohio State-Denison Conf., pp. 58–94. World Scientific, Singapore (1993)

    Google Scholar 

  6. Beidar, K., Wisbauer, R.: Properly semiprime self-pp-modules. Commun. Algebra 23, 841–861 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Birkenmeier, G.F., Lennon, M.J.: Extending sets of idempotents to ring extensions. Commun. Algebra (to appear)

    Google Scholar 

  8. Birkenmeier, G.F., Tercan, A.: When some complement of a submodule is a summand. Commun. Algebra 35, 597–611 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: On ring hulls. In: Rizvi, S.T., Zaidi, S.M.A. (eds.) Trends in Theory of Rings and Modules, pp. 17–34. Anamaya Pub., New Delhi (2005)

    Google Scholar 

  10. Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: Ring hulls of extension rings. In: Chen, J.L., Ding, N.Q., Marubayashi, H. (eds.) Advances in Ring Theory. Proc. of 4th China-Japan-Korea International Conference, pp. 12–25. World Scientific, Englewood Cliffs (2005)

    Chapter  Google Scholar 

  11. Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: Ring hulls and applications. J. Algebra 304, 633–665 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: Ring hulls of generalized triangular matrix rings. Aligarh Bull. Math. 25, 65–77 (2006)

    MathSciNet  Google Scholar 

  13. Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: Ring hulls of semiprime homomorphic images. In: Brzeziński, T., Gómez-Pardo, J.L., Shestakov, I., Smith, P.F. (eds.) Modules and Comodules. Trends in Math., pp. 101–111. Birkhäuser, Boston (2008)

    Chapter  Google Scholar 

  14. Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: The structure of rings of quotients. J. Algebra 321, 2545–2566 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: Hulls of semiprime rings with applications to C ∗-algebras. J. Algebra 322, 327–352 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: Modules with FI-extending hulls. Glasg. Math. J. 51, 347–357 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: Principally quasi-Baer ring hulls. In: Huynh, D.V., López-Permouth, S.R. (eds.) Advances in Ring Theory. Trends in Math., pp. 47–61. Birkhäuser, Boston (2010)

    Chapter  Google Scholar 

  18. Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: A theory of hulls for rings and modules. In: Albu, T., Birkenmeier, G.F., Erdogan, A., Tercan, A. (eds.) Ring and Module Theory. Trends in Math., pp. 27–71. Birkhäuser, Boston (2010)

    Chapter  Google Scholar 

  19. Burgess, W.D., Raphael, R.: On extensions of regular rings of finite index by central elements. In: Jain, S.K., Rizvi, S.T. (eds.) Advances in Ring Theory. Trends in Math., pp. 73–86. Birkhäuser, Boston (1997)

    Chapter  Google Scholar 

  20. Connell, I.G.: On the group rings. Can. J. Math. 15, 650–685 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  21. Curtis, C.W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. Interscience, New York (1962)

    MATH  Google Scholar 

  22. Divinsky, N.: Rings and Radicals. Univ. Toronto Press, Toronto (1965)

    MATH  Google Scholar 

  23. Dobbs, D.E., Picavet, G.: Weak Baer going-down rings. Houst. J. Math. 29, 559–581 (2003)

    MathSciNet  MATH  Google Scholar 

  24. Dorsey, T.J., Mesyan, Z.: On minimal extensions of rings. Commun. Algebra 37, 3463–3486 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Eckmann, B., Schopf, A.: Über injektive Moduln. Arch. Math. 4, 75–78 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  26. Faith, C.: Lectures on Injective Modules and Quotient Rings. Lecture Notes in Math., vol. 49. Springer, Berlin (1967)

    MATH  Google Scholar 

  27. Faith, C., Page, S.: FPF Ring Theory: Faithful Modules and Generators of Mod-R. Cambridge Univ. Press, Cambridge (1984)

    Book  Google Scholar 

  28. Faith, C., Utumi, Y.: Intrinsic extensions of rings. Pac. J. Math. 14, 505–512 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ferrero, M.: Closed submodules of normalizing bimodules over semiprime rings. Commun. Algebra 29, 1513–1550 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Fisher, J.W.: Structure of semiprime P.I. rings. Proc. Am. Math. Soc. 39, 465–467 (1973)

    MATH  Google Scholar 

  31. Fisher, J.W., Snider, R.L.: On the von Neumann regularity of rings with regular prime factor rings. Pac. J. Math. 54, 135–144 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  32. Fuchs, L.: Infinite Abelian Groups I. Academic Press, New York (1970)

    MATH  Google Scholar 

  33. Gardner, B.J., Wiegandt, R.: Radical Theory of Rings. Marcel Dekker, New York (2004)

    MATH  Google Scholar 

  34. Goel, V.K., Jain, S.K.: π-injective modules and rings whose cyclics are π-injective. Commun. Algebra 6, 59–73 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  35. Goodearl, K.R.: Ring Theory: Nonsingular Rings and Modules. Marcel Dekker, New York (1976)

    MATH  Google Scholar 

  36. Goodearl, K.R.: Von Neumann Regular Rings. Krieger, Malabar (1991)

    MATH  Google Scholar 

  37. Harada, M.: On the small hulls of a commutative ring. Osaka J. Math. 15, 679–682 (1978)

    MathSciNet  MATH  Google Scholar 

  38. Hirano, Y., Hongan, M., Ohori, M.: On right P.P. rings. Math. J. Okayama Univ. 24, 99–109 (1982)

    MathSciNet  MATH  Google Scholar 

  39. Jacobson, N.: Structure of Rings. Amer. Math. Soc. Colloq. Publ., vol. 37. Amer. Math. Soc., Providence (1964)

    Google Scholar 

  40. Jain, S.K., López-Permouth, S.R., Syed, S.R.: Mutual injective hulls. Can. Math. Bull. 39, 68–73 (1996)

    Article  MATH  Google Scholar 

  41. Jin, H.L., Doh, J., Park, J.K.: Quasi-Baer rings with essential prime radicals. Commun. Algebra 34, 3537–3541 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. Johnson, R.E.: Structure theory of faithful rings II, restricted rings. Trans. Am. Math. Soc. 84, 523–544 (1957)

    Google Scholar 

  43. Johnson, R.E., Wong, E.T.: Quasi-injective modules and irreducible rings. J. Lond. Math. Soc. 36, 260–268 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  44. Kaplansky, I.: Commutative Rings. Univ. Chicago Press, Chicago (1974)

    MATH  Google Scholar 

  45. Kist, J.: Minimal prime ideals in commutative semigroups. Proc. Lond. Math. Soc. 13, 31–50 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  46. Kulikov, L.Ya.: On the theory of Abelian group of arbitrary cardinality. Mat. Sb. 9, 165–182 (1941)

    Google Scholar 

  47. Kulikov, L.Ya.: On the theory of Abelian group of arbitrary cardinality. Mat. Sb. 16, 129–162 (1945)

    Google Scholar 

  48. Lam, T.Y.: Lectures on Modules and Rings. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  49. Lambek, J.: Lectures on Rings and Modules. Blaisdell, Waltham (1966)

    MATH  Google Scholar 

  50. Martindale, W.S. III: On semiprime P.I. rings. Proc. Am. Math. Soc. 40, 365–369 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  51. McCoy, N.H.: The Theory of Rings. Chelsea, New York (1973)

    MATH  Google Scholar 

  52. Mewborn, A.C.: Regular rings and Baer rings. Math. Z. 121, 211–219 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  53. Mohamed, S.H., Müller, B.J.: Continuous and Discrete Modules. Cambridge Univ. Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  54. Müller, B.J., Rizvi, S.T.: On the existence of continuous hulls. Commun. Algebra 10, 1819–1838 (1982)

    Article  MATH  Google Scholar 

  55. Oshiro, K.: On torsion free modules over regular rings. Math. J. Okayama Univ. 16, 107–114 (1973)

    MathSciNet  MATH  Google Scholar 

  56. Oshiro, K.: On torsion free modules over regular rings III. Math. J. Okayama Univ. 18, 43–56 (1975/76)

    MathSciNet  Google Scholar 

  57. Papp, Z.: On algebraically closed modules. Publ. Math. (Debr.) 6, 311–327 (1959)

    MathSciNet  MATH  Google Scholar 

  58. Passman, D.S.: Nil ideals in group rings. Mich. Math. J. 9, 375–384 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  59. Passman, D.S.: The Algebraic Structure of Group Rings. Wiley, New York (1977)

    MATH  Google Scholar 

  60. Raphael, R.M., Woods, R.G.: The epimorphic hull of C(X). Topol. Appl. 105, 65–88 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  61. Rizvi, S.T.: Contributions to the Theory of Continuous Modules. Doctoral Dissertation, McMaster Univ. (1981)

    Google Scholar 

  62. Rowen, L.H.: Polynomial Identities in Ring Theory. Academic Press, New York (1980)

    MATH  Google Scholar 

  63. Shoda, K.: Zur Theorie der algebraischen Erweiterungen. Osaka Math. J. 4, 133–143 (1952)

    MathSciNet  MATH  Google Scholar 

  64. Stenström, B.: Rings of Quotients. Springer, Berlin (1975)

    Book  MATH  Google Scholar 

  65. Storrer, H.: Epimorphismen von kommutativen Ringen. Comment. Math. Helv. 43, 378–401 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  66. Utumi, Y.: On quotient rings. Osaka Math. J. 8, 1–18 (1956)

    MathSciNet  MATH  Google Scholar 

  67. Utumi, Y.: On continuous regular rings and semisimple self injective rings. Can. J. Math. 12, 597–605 (1960)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, USA

    Gary F. Birkenmeier

  2. Department of Mathematics, Busan National University, Busan, South Korea

    Jae Keol Park

  3. Department of Mathematics, The Ohio State University at Lima, Lima, OH, USA

    S. Tariq Rizvi

Authors
  1. Gary F. Birkenmeier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jae Keol Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Tariq Rizvi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Birkenmeier, G.F., Park, J.K., Rizvi, S.T. (2013). Ring and Module Hulls. In: Extensions of Rings and Modules. Springer, New York, NY. https://doi.org/10.1007/978-0-387-92716-9_8

Download citation

Publish with us

Policies and ethics

Search

Navigation