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Baer, Rickart, and Quasi-Baer Rings

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Abstract

This chapter is devoted to properties of rings for which certain annihilators are direct summands. Such classes of rings include those of Baer rings, right Rickart rings, quasi-Baer rings, and right p.q.-Baer rings. The results and the material presented in this chapter will be instrumental in developing the subject of our study in later chapters. It is shown that the Baer and the Rickart properties of rings do not transfer to the rings of matrices or to the polynomial ring extensions, while the quasi-Baer and the p.q.-Baer properties of rings do so. The notions of Baer and Rickart rings are compared and contrasted in Sect. 3.1 and the notions of quasi-Baer and principally quasi-Baer rings in Sect. 3.2, respectively. A result of Chatters and Khuri shows that there are strong bonds between the extending and the Baer properties of rings. We shall also see some instances where the two notions coincide. It is shown that there are close connections between the FI-extending and the quasi-Baer properties for rings.

One of the motivations for the study of the quasi-Baer and p.q.-Baer rings is the fact that they behave better with respect to various extensions than the Baer and Rickart rings. For example, as shown in this chapter, each of the quasi-Baer and the p.q.-Baer properties is Morita invariant. This useful behavior will be effectively applied in later chapters. The results on the transference (or the lack of transference) of the properties presented in this chapter to matrix and polynomial ring extensions are intended to motivate further investigations on when these properties transfer to various other extensions.

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References

  1. Arens, R., Kaplansky, I.: Topological representation of algebras. Trans. Am. Math. Soc. 63, 457–481 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  2. Armendariz, E.P.: A note on extensions of Baer and p.p.-rings. J. Aust. Math. Soc. 18, 470–473 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  3. Armendariz, E.P., Berberian, S.K.: Baer rings satisfying J 2=J for all ideals. Commun. Algebra 17, 1739–1758 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Armendariz, E.P., Park, J.K.: Compressible matrix rings. Bull. Aust. Math. Soc. 30, 295–298 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Armendariz, E.P., Steinberg, S.A.: Regular self-injective rings with a polynomial identity. Trans. Am. Math. Soc. 190, 417–425 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  6. Armendariz, E.P., Koo, H.K., Park, J.K.: Compressible group algebras. Commun. Algebra 13, 1763–1777 (1985). Corrigendum 215, 99–100 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Armendariz, E.P., Birkenmeier, G.F., Park, J.K.: Ideal intrinsic extensions with connections to PI-rings. J. Pure Appl. Algebra 213, 1756–1776 (2009). Corrigendum 215, 99–100 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Baer, R.: Linear Algebra and Projective Geometry. Academic Press, New York (1952)

    MATH  Google Scholar 

  9. Berberian, S.K.: Baer ∗-Rings. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  10. Berberian, S.K.: The center of a corner of a ring. J. Algebra 71, 515–523 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. Berberian, S.K.: Baer Rings and Baer ∗-Rings. University of Texas, Austin (1991)

    Google Scholar 

  12. Bergman, G.M.: Hereditary commutative rings and centres of hereditary rings. Proc. Lond. Math. Soc. 23, 214–236 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bergman, G.M.: Some examples of non-compressible rings. Commun. Algebra 12, 1–8 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  14. Birkenmeier, G.F.: A Decomposition Theory of Rings. Doctoral Dissertation, University of Wisconsin at Milwaukee (1975)

    Google Scholar 

  15. Birkenmeier, G.F.: Baer rings and quasi-continuous rings have a MDSN. Pac. J. Math. 97, 283–292 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  16. Birkenmeier, G.F.: A generalization of FPF rings. Commun. Algebra 17, 855–884 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Birkenmeier, G.F.: Decompositions of Baer-like rings. Acta Math. Hung. 59, 319–326 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Birkenmeier, G.F., Huang, F.K.: Annihilator conditions on polynomials. Commun. Algebra 29, 2097–2112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Birkenmeier, G.F., Huang, F.K.: Annihilator conditions on formal power series. Algebra Colloq. 9, 29–37 (2002)

    MathSciNet  MATH  Google Scholar 

  20. Birkenmeier, G.F., Huang, F.K.: Annihilator conditions on polynomials II. Monatshefte Math. 141, 265–276 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Birkenmeier, G.F., Kim, J.Y., Park, J.K.: When is the CS condition hereditary? Commun. Algebra 27, 3875–3885 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Birkenmeier, G.F., Heatherly, H.E., Kim, J.Y., Park, J.K.: Triangular matrix representations. J. Algebra 230, 558–595 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Birkenmeier, G.F., Kim, J.Y., Park, J.K.: Quasi-Baer ring extensions and biregular rings. Bull. Aust. Math. Soc. 61, 39–52 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Birkenmeier, G.F., Kim, J.Y., Park, J.K.: Rings with countably many direct summands. Commun. Algebra 28, 757–769 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Birkenmeier, G.F., Kim, J.Y., Park, J.K.: Principally quasi-Baer rings. Commun. Algebra 29, 639–660 (2001). Erratum 30, 5609 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Birkenmeier, G.F., Kim, J.Y., Park, J.K.: Triangular matrix representations of semiprimary rings. J. Algebra Appl. 2, 123–131 (2002)

    Article  MathSciNet  Google Scholar 

  27. Birkenmeier, G.F., Müller, B.J., Rizvi, S.T.: Modules in which every fully invariant submodule is essential in a direct summand. Commun. Algebra 30, 1395–1415 (2002)

    Article  MATH  Google Scholar 

  28. Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: Modules with fully invariant submodules essential in fully invariant summands. Commun. Algebra 30, 1833–1852 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: Generalized triangular matrix rings and the fully invariant extending property. Rocky Mt. J. Math. 32, 1299–1319 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. 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 

  32. Birkenmeier, G.F., Kim, J.Y., Park, J.K.: The factor ring of a quasi-Baer ring by its prime radical. J. Algebra Appl. 10, 157–165 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Brown, K.A.: The singular ideals of group rings. Q. J. Math. Oxf. 28, 41–60 (1977)

    Article  MATH  Google Scholar 

  34. Castella, D.: Anneaux régulaiers de Baer compressibles. Commun. Algebra 15, 1621–1635 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  35. Chase, S.U.: A generalization of the ring of triangular matrices. Nagoya Math. J. 18, 13–25 (1961)

    MathSciNet  MATH  Google Scholar 

  36. Chatters, A.W., Hajarnavis, C.R.: Rings with Chain Conditions. Pitman, Boston (1980)

    MATH  Google Scholar 

  37. Chatters, A.W., Khuri, S.M.: Endomorphism rings of modules over non-singular CS rings. J. Lond. Math. Soc. 21, 434–444 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  38. Clark, W.E.: Baer rings which arise from certain transitive graphs. Duke Math. J. 33, 647–656 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  39. Clark, W.E.: Twisted matrix units semigroup algebras. Duke Math. J. 34, 417–424 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  40. Contessa, M.: A note on Baer rings. J. Algebra 118, 20–32 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  41. DeMeyer, F., Ingraham, E.: Separable Algebras over Commutative Rings. Lecture Notes in Math., vol. 181. Springer, Berlin (1971)

    MATH  Google Scholar 

  42. Endo, S.: Note on p.p. rings (a supplement to Hattori’s paper). Nagoya Math. J. 17, 167–170 (1960)

    MathSciNet  MATH  Google Scholar 

  43. Evans, M.W.: On commutative P.P. rings. Pac. J. Math. 41, 687–697 (1972)

    Article  MATH  Google Scholar 

  44. Faith, C.: Injective quotient rings of commutative rings. In: Module Theory. Lecture Notes in Math., vol. 700, pp. 151–203. Springer, Berlin (1979)

    Chapter  Google Scholar 

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

    Book  Google Scholar 

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

    MATH  Google Scholar 

  47. Goodearl, K.R.: Simple Noetherian rings—the Zalesskii-Neroslavskii examples. In: Handelman, D., Lawrence, J. (eds.) Ring Theory, Waterloo, 1978. Lecture Notes in Math., vol. 734, pp. 118–130. Springer, Berlin (1979)

    Google Scholar 

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

    MATH  Google Scholar 

  49. Han, J., Hirano, Y., Kim, H.: Semiprime Ore extensions. Commun. Algebra 28, 3795–3801 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  50. Hattori, A.: A foundation of torsion theory for modules over general rings. Nagoya Math. J. 17, 147–158 (1960)

    MathSciNet  MATH  Google Scholar 

  51. Holland, S.S. Jr.: Remarks on type I Baer and Baer ∗-rings. J. Algebra 27, 516–522 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  53. Jeremy, L.: L’arithmétique de von Neumann pour les facteurs injectifs. J. Algebra 62, 154–169 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  54. Jøndrup, S.: p.p. rings and finitely generated flat ideals. Proc. Am. Math. Soc. 28, 431–435 (1971)

    Google Scholar 

  55. Kaplansky, I.: Rings of operators. Univ. Chicago Mimeographed Lecture Notes (Notes by S.K. Berberian, with an Appendix by R. Blattner), Univ. Chicago (1955)

    Google Scholar 

  56. Kaplansky, I.: Rings of Operators. Benjamin, New York (1968)

    MATH  Google Scholar 

  57. Kim, J.Y., Park, J.K.: When is a regular ring a semisimple Artinian ring? Math. Jpn. 45, 311–313 (1997)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  59. Lawrence, J.: A singular primitive ring. Proc. Am. Math. Soc. 45, 59–62 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  60. Lenzing, H.: Halberbliche Endomorphismenringe. Math. Z. 118, 219–240 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  61. Liu, Z.: Baer rings of generalized power series. Glasg. Math. J. 44, 463–469 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  62. Liu, Z., Zhao, R.: A generalization of pp-rings and p.q.-Baer rings. Glasg. Math. J. 48, 217–229 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  63. Maeda, S.: On a ring whose principal right ideals generated by idempotents form a lattice. J. Sci. Hiroshima Univ. Ser. A 24, 509–525 (1960)

    MathSciNet  MATH  Google Scholar 

  64. Martindale, W.S. III: Prime rings satisfying a generalized polynomial identity. J. Algebra 12, 576–584 (1969)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  66. Nicholson, W.K.: On PP-rings. Period. Math. Hung. 27, 85–88 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  67. Osofsky, B.L.: A non-trivial ring with non-rational injective hull. Can. Math. Bull. 10, 275–282 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  68. Pollingher, A., Zaks, A.: On Baer and quasi-Baer rings. Duke Math. J. 37, 127–138 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  69. Posner, E.C.: Prime rings satisfying a polynomial identity. Proc. Am. Math. Soc. 11, 180–183 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  70. Rangaswamy, K.M.: Representing Baer rings as endomorphism rings. Math. Ann. 190, 167–176 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  71. Rangaswamy, K.M.: Regular and Baer rings. Proc. Am. Math. Soc. 42, 354–358 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  72. Rizvi, S.T., Roman, C.S.: Baer and quasi-Baer modules. Commun. Algebra 32, 103–123 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  73. Rowen, L.H.: Some results on the center of a ring with polynomial identity. Bull. Am. Math. Soc. 79, 219–223 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  74. Rowen, L.H.: Maximal quotients of semiprime PI-algebras. Trans. Am. Math. Soc. 196, 127–135 (1974)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  76. Small, L.W.: Semihereditary rings. Bull. Am. Math. Soc. 73, 656–658 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  77. Speed, T.P.: A note on commutative Baer rings II. J. Aust. Math. Soc. 14, 257–263 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  78. Speed, T.P.: A note on commutative Baer rings III. J. Aust. Math. Soc. 15, 15–21 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  79. Speed, T.P., Evans, M.W.: A note on commutative Baer rings. J. Aust. Math. Soc. 12, 1–6 (1972)

    Article  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  81. Stone, M.H.: Applications of the theory of Boolean rings to general topology. Trans. Am. Math. Soc. 41, 375–481 (1937)

    Article  Google Scholar 

  82. Xue, W.: On PP-rings. Kobe Math. J. 7, 77–80 (1990)

    MATH  Google Scholar 

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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

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  1. Gary F. Birkenmeier
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  2. Jae Keol Park
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Birkenmeier, G.F., Park, J.K., Rizvi, S.T. (2013). Baer, Rickart, and Quasi-Baer Rings. In: Extensions of Rings and Modules. Springer, New York, NY. https://doi.org/10.1007/978-0-387-92716-9_3

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