Semiprimitive Rings, Semiprime Rings, and the Nil Radical

  • Carl Faith
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 191)


A ring R is semiprime (semiprimitive) if and only the intersection of the prime (primitive) ideals is zero. Then, R is a subdirect product of prime (primitive) rings 26.6 and 26.13. The (McCoy) prime radical of a ring is defined to be the intersection of the prime ideals, and is characterized as the set of all strongly nilpotent elements of R (theorem of Levitzki 26.5). When R is commutative, this is just the set of nilpotent elements.


Prime Ideal Prime Ring Nilpotent Element Subdirect Product Semiprime Ring 
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  1. [67]
    Armendariz, E. P.: On radical extensions of rings. J. Austral. Math. Soc. 7, 552–554 (1967).MathSciNetzbMATHCrossRefGoogle Scholar
  2. [56a]
    Amitsur, S.A.: Algebras over infinite fields. Proc. Amer. Math. Soc. 7, 35–48 (1956).MathSciNetzbMATHCrossRefGoogle Scholar
  3. [56b]
    Amitsur, S.A.: Radical of polynomial rings. Canad. J. Math. 8, 355–361 (1956).MathSciNetzbMATHCrossRefGoogle Scholar
  4. [57a]
    Amitsur, S.A.: Derivations in simple rings. Proc. Lond. Math. Soc. (3) 7, 87–112 (1957).MathSciNetzbMATHCrossRefGoogle Scholar
  5. [57b]
    Amitsur, S.A.: A generalization of Hilbert’s Nullstellensatz. Proc. Amer. Math. Soc. 8, 649–656 (1957).MathSciNetzbMATHGoogle Scholar
  6. [57c]
    Amitsur, S. A.: The radical of field extensions. Bull. Res. Council of Israel (Israel J. Math.) 7F, 1–10 (1957).MathSciNetGoogle Scholar
  7. [59]
    Amitsur, S.A.: On the semi-simplicity of group algebras. Mich. Math. J. 6, 251–253 (1956).MathSciNetGoogle Scholar
  8. [73a]
    Amitsur, S.A.: Nil radicals. Historical notes and some new results, published in the book edited by Kertész [73].Google Scholar
  9. [73b]
    Amitsur, S.A.: Polynomial identities and Azumaya algebras. J. Algebra 27, 117–125 (1973).MathSciNetzbMATHCrossRefGoogle Scholar
  10. [43a]
    Baer, R.: Radical ideals. Amer. J. Math. 65, 537–568 (1943).MathSciNetzbMATHCrossRefGoogle Scholar
  11. [64]
    Bergman, G.M.: A ring primitive on the right but not the left. Proc. Amer. Math. Soc. 15, 473–475 (1964).MathSciNetzbMATHCrossRefGoogle Scholar
  12. [7]
    Brenner [7]Google Scholar
  13. [47]
    Brown, B., McCoy, N.: Radicals and subdirect sums. Amer. J. Math. 69, 46–58 (1947).MathSciNetzbMATHCrossRefGoogle Scholar
  14. [48]
    Brown, B., McCoy, N.: The radical of a ring. Duke Math. J. 15, 495–499 (1948).MathSciNetzbMATHCrossRefGoogle Scholar
  15. [54]
    Curtis [54]Google Scholar
  16. [69]
    Eckstein, F.: On the Mal’cev theorem. J. Algebra 12, 372–385 (1969).MathSciNetzbMATHCrossRefGoogle Scholar
  17. [60]
    Faith, C.: Algebraic division ring extensions. Proc. Amer. Math. Soc. 11, 43–53 (1960).MathSciNetzbMATHCrossRefGoogle Scholar
  18. [62]
    Faith, C.: Strongly regular extensions of rings. Nagoya Math. J. 20, 169–183 (1962).MathSciNetGoogle Scholar
  19. [35a]
    Fitting, H.: Über die direkten Produktzerlegungen einer Gruppe in direkt unzerlegbare Faktoren. Math. Z. 39, 19–41 (1935).MathSciNetCrossRefGoogle Scholar
  20. [35b]
    Fitting, H.: Primärkomponentenzerlegung in nichtkommutativen Ringen. Math. Ann. 111, 19–41 (1935).MathSciNetCrossRefGoogle Scholar
  21. [51]
    Goldman, O.: Hilbert rings, and the Hilbert Nullstellensatz. Math. Z. 54, 136–140 (1951).MathSciNetzbMATHCrossRefGoogle Scholar
  22. [72]
    Hampton, C.R., Passman, D.S.: On the semisimplicity of group rings of solvable groups. Trans. Amer. Math. Soc. 173, 289–301 (1972).MathSciNetzbMATHCrossRefGoogle Scholar
  23. [53]
    Herstein, I.N.: Finite multiplicative subgroups in division rings. Pac. J. Math. 1, 121–126 (1953).MathSciNetGoogle Scholar
  24. [43]
    Jacobson, N.: The theory of rings, Surveys of the Amer. Math. Soc. vol. 2, Providence 1942.Google Scholar
  25. [45a]
    Jacobson, N.: The radical and semisimplicity for arbitrary rings. Amer. J. Math. 67, 300–342 (1945).MathSciNetzbMATHCrossRefGoogle Scholar
  26. [45b]
    Jacobson, N.: The structure of simple rings without finiteness assumptions. Trans. Amer. Math. Soc. 57, 228–245 (1945).MathSciNetzbMATHCrossRefGoogle Scholar
  27. [45c]
    Jacobson, N.: Structure theory for algebraic algebras of bounded degree. Ann. of Math. 46, 695–707 (1945).MathSciNetzbMATHCrossRefGoogle Scholar
  28. [55, 64]
    Jacobson, N.: Structure of Rings, Revised. Colloquium Publication, vol.37. Amer. Math. Soc, Providence 1955, 1964.Google Scholar
  29. [68, 71]
    JategaonkarGoogle Scholar
  30. [30a]
    Köthe, G.: Die Struktur der Ringe deren Restklassenring nach dem Radical vollständig reduzibel ist. Math. Z. 32, 161–186 (1930).MathSciNetzbMATHCrossRefGoogle Scholar
  31. [30b]
    Köthe, G.: Über maximale nilpotente Unterringe und Nilringe. Math. Ann. 103, 359–363 (1938).CrossRefGoogle Scholar
  32. [28]
    Krull, W.: Zur Theorie der Allgemeinen Zahlringe. Math. Ann. 99, 51–70 (1928).MathSciNetzbMATHCrossRefGoogle Scholar
  33. [48]
    Azumaya, G.: On generalized semiprimary rings and Krull-Remak-Schmidt’s theorem. Japan J. Math. 19, 525–647 (1948).MathSciNetzbMATHGoogle Scholar
  34. [51]
    Azumaya, G.: On maximally central algebras. Nagoya Math. J. 2, 119–150 (1951).MathSciNetzbMATHGoogle Scholar
  35. [66]
    Lambek, J.: Rings and Modules. Blaisdell, New York 1966.zbMATHGoogle Scholar
  36. [73]
    Lenagen [73]Google Scholar
  37. [51]
    Levitzki, J.: Prime ideals and the lower radical. Amer. J. Math. 73, 25–29 (1951).MathSciNetzbMATHCrossRefGoogle Scholar
  38. [42]
    Mal’cev, A. I.: On the representation of an algebra as a direct sum of its radical and a semi-simple algebra. Dokl. Akad. Nauk. SSSR 36, 42–45 (1942).MathSciNetGoogle Scholar
  39. [49]
    McCoy, N.H.: Prime ideals in general rings. Amer. J. Math. 71, 823–833 (1949).MathSciNetzbMATHCrossRefGoogle Scholar
  40. [55]
    McCoy, N. H.: Subdirect sum representations of prime rings. Duke Math. J. 22, 357–364 (1955).MathSciNetzbMATHCrossRefGoogle Scholar
  41. [56]
    McCoy, N.H.: The prime radical of a polanomial ring. Publ. Math. Debrecen 4, 161–162 (1956).MathSciNetzbMATHGoogle Scholar
  42. [57a]
    McCoy, N.H.: A note on finite unions of ideals and subgroups. Proc. Amer. Math. Soc. 8, 633–637 (1957).MathSciNetzbMATHCrossRefGoogle Scholar
  43. [64]
    McCoy, N. H.: Theory of Rings. McMillan, New York 1964.zbMATHGoogle Scholar
  44. [56]
    Mostow, G.D.: Fully reducible subgroups of algebraic groups. Amer. J. Math. 78, 200–221 (1956).MathSciNetzbMATHCrossRefGoogle Scholar
  45. [61]
    Patterson [61]Google Scholar
  46. [72]
    Shock [72]Google Scholar
  47. [50]
    Snapper [50]Google Scholar
  48. [57]
    Taft, E. J.: Invariant Wedderburn factors. Ill. J. Math. 1, 565–573 (1957).MathSciNetzbMATHGoogle Scholar
  49. [64]
    Taft, E.J.: Orthogonal conjugates in associative and Lie algebras. Trans. Amer. Math. Soc. 113, 18–29 (1964).MathSciNetzbMATHCrossRefGoogle Scholar
  50. [68]
    Taft, E. J.: Cohomology of groups of algebra automorphisms. J. Algebra 10, 400–410 (1968).MathSciNetzbMATHCrossRefGoogle Scholar
  51. [05]
    WedderburnGoogle Scholar

Additional References

  1. [54, 61, 67, 73 a]
    Amitsur [54, 61, 67, 73 a]Google Scholar
  2. [66]
    Amitsur and ProcesiGoogle Scholar
  3. [71b]
    Beachy, J.A.: On quasi-Artinian rings. J. Lond. Math. Soc. (2) 3, 449–452 (1971).MathSciNetzbMATHCrossRefGoogle Scholar
  4. [73]
    Borho, W., Gabriel, P., Rentschler, R.: Primideale in einhüllenden auslösbaren Lie-Algebren. Lecture Notes in Mathematics, vol. 357. New York-Heidelberg-Berlin: Springer 1973.Google Scholar
  5. [76]
    Cauchon, G.: Les T-anneaux, la condition (H) de Gabriel et ses conséquences. Comm. Algebra 4, 1–10 (1976).MathSciNetCrossRefGoogle Scholar
  6. [68]
    Dixmier, J.: Sur les algèbres de Weyl. Bull. Soc. Math. France 96, 209–242 (1968).MathSciNetzbMATHGoogle Scholar
  7. [72b]
    Formanek, E.: A problem of Passman on semisimplicity. Bull. Lond. Math. Soc. 4, 375–376 (1972).MathSciNetzbMATHCrossRefGoogle Scholar
  8. [73a]
    Formanek, E.: Group rings of free products are primitive. J. Algebra 26, 508–511 (1973).MathSciNetzbMATHCrossRefGoogle Scholar
  9. [72]
    Formanek, E, Snider, R.L.: Primitive group rings. Proc. Amer. Math. Soc. 36, 375–376 (1972).MathSciNetCrossRefGoogle Scholar
  10. [65]
    Fluch, W.: Gruppen ohne endlich-dimensionale Darstellungen. Math. Scand. 16, 164–168 (1965).MathSciNetzbMATHGoogle Scholar
  11. [74a]
    Jategaonkar, A. V.: Jacobson’s conjecture and modules over fully bounded noetherian rings. J. Algebra 30, 103–121 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  12. [75]
    Jategaonkar, A. V.: Principal ideal theorem for Noetherian P. I. rings.Google Scholar
  13. [48]
    Kaplansky, I.: Rings with polynomial identity. Bull. Amer. Math. Soc. 54, 575–580 (1948).MathSciNetzbMATHCrossRefGoogle Scholar
  14. [49]
    Kaplansky, I.: Elementary divisors and modules. Trans. Amer. Math. Soc. 66, 464–491 (1949).MathSciNetzbMATHCrossRefGoogle Scholar
  15. [68]
    Kaplansky, I.: Commutative rings. Proceedings of the Canadian Mathematical Congress, Manitoba 1968.Google Scholar
  16. [53]
    Kurosch, A.: Radicals of rings and algebras. Mat. Sbornik 33, 13–26 (1953).MathSciNetGoogle Scholar
  17. [74]
    Lawrence, J.: A singula primitive ring. Proc. Amer. Math. Soc. 45, 59–62 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  18. [75a]
    Lawrence, J.: The coefficient ring of a primitive group ring. Canad. J. Math. 27, 489–494 (1975).MathSciNetzbMATHCrossRefGoogle Scholar
  19. [72]
    Mitchell, B.: Rings with several objects. Advances in Math. 8, 1–161 (1972).MathSciNetzbMATHCrossRefGoogle Scholar
  20. [71]
    Passman, D.S.: Infinite group rings. Marcel Dekker, New York 1971.zbMATHGoogle Scholar
  21. [73]
    Rosenblade, J.E.: Group rings of polycyclic groups. J. Pure and Applied Algebra 3, 307–328 (1973).MathSciNetCrossRefGoogle Scholar
  22. [73]
    Small, L. W.: Prime ideals in Noetherian Pi-rings. Bull. Amer. Math. Soc. 79, 421–422 (1973).MathSciNetzbMATHCrossRefGoogle Scholar
  23. [71]
    Smith, M.: Group algebras. J. Algebra 18, 477–499 (1971).MathSciNetzbMATHCrossRefGoogle Scholar
  24. [74]
    Snider [74]Google Scholar
  25. [68, 75]
    Varnos [68, 75].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1976

Authors and Affiliations

  • Carl Faith
    • 1
    • 2
  1. 1.Rutgers, The State UniversityNew BrunswickUSA
  2. 2.The Institute for Advanced StudyPrincetonUSA

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