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Semilocal Rings and the Jacobson Radical

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Book cover Algebra II Ring Theory

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 191))

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Abstract

The main topics of this chapter are (1) the Jacobson radical of a ring and of a module 18.0; (2) the Perlis-Jacobson characterization 18.6 of the radical of a ring; (3) local rings 18.10; (4) semiprimary rings 18.12; (5) the theorem of Hopkins and Levitzki 18.13; (6) the Krull-Schmidt or Unique Decomposition Theorem 18.18; (7) the basic module and ring 18.21–23; (8) the Chinese remainder theorem 18.30–32 and primary decomposable rings 18.36–37; and (9) the characterization 18.47 of rings with semilocal right quorings. (As defined by (I, p. 4.10), R is semilocal if R/rad R is semisimple. Also see 18.10A.)

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References

  1. Amitsur, S.A.: Algebras over infinite fields. Proc. Amer. Math. Soc. 7, 35–48 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  2. Amitsur, S.A.: Radical of polynomial rings. Canad. J. Math. 8, 355–361 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  3. Asano, K.: Über Hauptidealringe mit Kettensatz. Osaka Math. J. 1, 52–61 (1949).

    MathSciNet  MATH  Google Scholar 

  4. Asano, K.: Über die Quotientenbildung von Schiefringen. J. Math. 2, 73–79 (1949).

    MathSciNet  Google Scholar 

  5. Azumaya, G.: Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt’s theorem. Nagoya Math. J. 1, 117–124 (1950).

    MathSciNet  MATH  Google Scholar 

  6. Baer, R.: Radical ideals. Amer. J. Math. 65, 537–568 (1943).

    Article  MathSciNet  MATH  Google Scholar 

  7. Baer, R.: Rings with duals. Amer. J. Math. 65, 569–584 (1943).

    Article  MathSciNet  MATH  Google Scholar 

  8. Bass, H.: Finitistic dimension and a homological generalization of semiprimary rings. Trans. Amer. Math. Soc. 95, 466–488 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  9. Bass, H.: Algebraic X-theory. Benjamin, N. Y., and Amsterdam 1968.

    Google Scholar 

  10. Björk, J.E.: Rings satisfying a minimum condition on principal ideals. J. reine u. angew. Math. 236, 466–488 (1969).

    Google Scholar 

  11. Björk, J.E.: On subrings of matrix rings over fields. Proc. Cam. Phil. Soc. 68, 275–284 (1970).

    Article  MATH  Google Scholar 

  12. Björk, J.E.: Rings satisfying certain chain conditions. J. reine u. angew. Math. 237 (1970).

    Google Scholar 

  13. Bourbaki, N.: Eléments de Mathématique (A. S. I. N° 1314), Algèbre Commutative, Chapitre 7 (Diviseurs). Hermann, Paris 1965.

    Google Scholar 

  14. Boyle, A.K.: When projective covers and injective hulls are isomorphic. Bull. Austral. Math. Soc. 8, 471–476 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  15. Chase, S.U.: Direct products of modules. Trans. Amer. Math. Soc. 97, 457–473 (1960).

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Cohen, I.S.: Commutative rings with restricted minimum condition. Duke Math. J. 17, 27–42 (1950).

    Article  MathSciNet  MATH  Google Scholar 

  18. Eilenberg, Nagao, and Nakayama

    Google Scholar 

  19. Elizarov, V.P.: Quotient rings. Algebra and Logic 8, 219–243 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  20. Faith, C.: Orders in semilocal rings. Bull. Amer. Math. Soc. 77, 960–962 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  21. Faith and Utumi

    Google Scholar 

  22. Feller, E.H., Swokowski, E.W.: Reflective N-prime rings with the ascending condition. Trans. Amer. Math. Soc. 99, 264–271 (1961); corrections, ibid. p. 555.

    MathSciNet  MATH  Google Scholar 

  23. Feller, E.H., Swokowski, E. W.: Reflective rings with the ascending chain condition. Proc. Amer. Math. Soc. 12, 651–653 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  24. Fitting, H.: Die Theorie der Automorphismenringe Abelscher Gruppen und ihr Analogon bei nicht kommutativen Gruppen. Math. Ann. 107, 514–542 (1933).

    Article  MathSciNet  Google Scholar 

  25. Fuller, K.R.: On indecomposable injectives over Artinian rings. Pac. J. Math. 29, 115–135 (1969).

    MathSciNet  MATH  Google Scholar 

  26. Harada, M.: Hereditary orders. Trans. Amer. Math. Soc. 107, 273–290 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  27. Harada, M.: Multiplicative ideal theory in hereditary orders. J. Math. Osaka City U. 14, 83–106 (1963).

    MathSciNet  Google Scholar 

  28. Harada, M.: Hereditary semiprimary rings and triangular matrix rings. Nagoya J. Math. 27, 463–484 (1966).

    MathSciNet  MATH  Google Scholar 

  29. Hopkins, C.: Rings with minimal condition for left ideals. Ann. of Math. 40, 712–730 (18–18) (1939).

    Article  MathSciNet  Google Scholar 

  30. Huppert, B.: Endliche Gruppen I. Die Grundlehren der math. Wiss. in Einzeldarstellungen, Bd. 134. Springer, Berlin-Heidelberg-New York 1967.

    Google Scholar 

  31. Jacobson, N.: The theory of rings, Surveys of the Amer. Math. Soc. vol. 2, Providence 1942.

    Google Scholar 

  32. Jacobson, N.: The radical and semisimplicity for arbitrary rings. Amer. J. Math. 67, 300–342 (1945).

    Article  MathSciNet  MATH  Google Scholar 

  33. Jacobson, N.: The structure of simple rings without finiteness assumptions. Trans. Amer. Math. Soc. 57, 228–245 (1945).

    Article  MathSciNet  MATH  Google Scholar 

  34. Jans and Nakayama

    Google Scholar 

  35. Kaplansky, I.: On a problem of Kurosch and Jacobson. Bull. Amer. Soc. 52, 496–500 (1946).

    Article  MathSciNet  MATH  Google Scholar 

  36. Kaplansky, I.: Commutative rings. Proceedings of the Canadian Mathematical Congress, Manitoba 1968.

    Google Scholar 

  37. Köthe, G.: Die Struktur der Ringe deren Restklassenring nach dem Radical vollständig reduzibel ist. Math. Z. 32, 161–186 (1930).

    Article  MathSciNet  MATH  Google Scholar 

  38. Köthe, G.: Über maximale nilpotente Unterringe und Nilringe. Math. Ann. 103, 359–363 (1938).

    Article  Google Scholar 

  39. Krull, W.: Über verallgemeinerte endliche Abelsche Gruppen. Math. Z. 23, 161–196 (1925).

    Article  MathSciNet  MATH  Google Scholar 

  40. Krull, W.: Dimensionstheorie in Stellenringen. J. reine u. angew. Math. 179, 204–226 (1938).

    Article  Google Scholar 

  41. Krull, W.: Idealtheorie. Chelsea, New York 1948.

    Google Scholar 

  42. Levitzki, J.: On rings which satisfy the minimum condition for right-hand ideals. Compositio Math. 7, 214–222 (1939).

    MathSciNet  MATH  Google Scholar 

  43. Levitzki, J.: A characteristic condition for semiprimary rings. Duke Math. J. 11, 367–368 (1944).

    Article  MathSciNet  MATH  Google Scholar 

  44. Levitzki, J.: Solution of a problem of G.Koethe. Amer. J. Math. 67, 437–442 (1945).

    Article  MathSciNet  MATH  Google Scholar 

  45. Nagata, M.: On the theory of semilocal rings. Proc. Japan Acad. 26, 131–140 (1950).

    Article  MathSciNet  Google Scholar 

  46. Nagata, M.: Note some studies on semilocal rings. Nagoya Math. J. 2, 23–30 (1951).

    Google Scholar 

  47. Nagata, M.: Note on subdirect sums of rings. Nagoya Math. J. 2, 49–53 (1951).

    MathSciNet  MATH  Google Scholar 

  48. Nagata, M.: On the fourteenth problem of Hilbert. Proc. Internat. Congress of Math., Cambridge U. Press 1958.

    Google Scholar 

  49. Nagata, M.: Local Rings. Interscience, New York 1962.

    MATH  Google Scholar 

  50. Nakayama, T.: Note on uniserial and generalized uniserial rings. Proc. Imp. Acad. Tokyo 16, 285–289 (1940).

    Article  MathSciNet  Google Scholar 

  51. Nakayama, T.: Algebras with antiisomorphic left and right ideal lattices. Proc. Imp. Acad. Tokyo 17, 53–56 (1940).

    Article  MathSciNet  Google Scholar 

  52. Noether, E.: Hyperkomplexe Größen und Darstellungstheorie. Math. Z. 30, 641–692 (1929).

    Article  MathSciNet  MATH  Google Scholar 

  53. Ornstein, A. J.: Rings with restricted minimum condition. Proc. Amer. Math. Soc. 19, 1145–1150 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  54. Perlis

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  56. Posner, E.C.: Differentiably simple rings. Proc. Amer. Math. Soc. 11, 337–343 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  57. Robson, J.C.: Artinian quotient rings. Proc. Lond. Math. Soc. 17, 600–616 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  58. Robson, J.C.: Pri-rings, and Ipri-rings. Quarterly J. Math. 18, 125–145 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  59. Schmidt, O.: Über unendliche Gruppen mit endlicher Kette. Math. Z. 29, 34–41 (1928).

    Article  MATH  Google Scholar 

  60. Schreier, O.: Über den Jordan-Hölderschen Satz. Abh. Math. Sem. U. Hamburg 6, 300–302 (1928).

    Article  MATH  Google Scholar 

  61. Swan, R.G.: Algebraic K-Theory. Lecture Notes in Mathematics, vol.76. Springer, Berlin-Heidelberg-New York 1968.

    Google Scholar 

  62. Zaks, A.: Semiprimary rings of generalized triangular type. J. Algebra 9, 54–78 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  63. Zelmanowitz, J. M.: Endomorphism rings of torsion-less modules. J. Algebra 5, 325–341 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  64. Greenberg, B.: Global dimension of cartesian squares. J. Algebra 32, 31–43 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  65. Vasconcelos, W.V.: The local rings of global dimension two. Proc. Amer. Math. Soc. 35, 381–386 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  66. Artin, E.: Zur Theorie der hyperkomplexen Zahlen. Abh. Math. Sem. U. Hamburg 5, 251–260 (1927).

    Article  MATH  Google Scholar 

  67. Artin, E.: Zur Arithmetik hyperkomplexer Zahlen. Abh. Math. Sem. U. Hamburg 5, 261–289 (1927).

    Article  MATH  Google Scholar 

  68. Ginn, S. M., Moss, P. B.: Finitely embedded modules over Noetherian rings. Bull. Amer. Math. Soc. 81, 709–710 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  69. Jacobinski, H.: Two remarks about hereditary orders. Proc. Amer. Math. Soc. 28, 1–8 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  70. Michler, G. O.: On maximal nilpotent subrings of right Noetherian rings. Glasgow Math.J. 8, 89–101 (1966).

    Article  MathSciNet  Google Scholar 

  71. Morita, K.: Duality for modules and its applications to the theory of rings with minimum condition. Sci Rpts. Tokyo Kyoiku Daigaku 6, 83–142 (1958).

    MATH  Google Scholar 

  72. Reiner.

    Google Scholar 

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Authors and Affiliations

  1. Rutgers, The State University, 08903, New Brunswick, N.J., USA

    Carl Faith

  2. The Institute for Advanced Study, 08540, Princeton, N.J., USA

    Carl Faith

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Editors and Affiliations

  1. Rutgers, The State University, 08903, New Brunswick, N.J., USA

    Carl Faith

  2. The Institute for Advanced Study, 08540, Princeton, N.J., USA

    Carl Faith

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© 1976 Springer-Verlag Berlin Heidelberg

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Faith, C. (1976). Semilocal Rings and the Jacobson Radical. In: Faith, C. (eds) Algebra II Ring Theory. Grundlehren der mathematischen Wissenschaften, vol 191. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-65321-6_3

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  • DOI: https://doi.org/10.1007/978-3-642-65321-6_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-65323-0

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