Algebra pp 419-442 | Cite as

Tensor Products and Flat Modules

  • Carl Faith
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 190)

Abstract

Let R be a ring, M a right and N a left R-module. As usual, M × N denotes cartesian product. If G is an abelian group, and g: M× NG is a mapping of sets, then for each yN there is a mapping g y : MG, defined by the formula g y (α) = g (α, y) ∀ αM. Symmetrically, if xM, then g x : NG is defined by the formula g x (b) = g (x, b) ∀bM. A mapping g:M×NG is bilinear in case g y : MG and g x : NG are group homomorphisms ∀ x M,yN. A balanced map g:M×NG is a bilinear map such that
$$g\left( {xr,y} \right) = g\left( {x,ry} \right)$$
x M, yN, rR.

Keywords

Proal Prool 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [57]
    Artin, E.: Geometric Algebra. New York: Interscience Publishers 1957.Google Scholar
  2. [57]
    Auslander, M.: On regular group rings. Proc. Amer. Math. Soc. 8, 658–664 (1957).MATHMathSciNetCrossRefGoogle Scholar
  3. [57]
    Auslander, M., Buchsbaum, D.: Homological dimension in local rings. Trans. Amer. Math. Soc. 85, 390–405 (1957).MATHMathSciNetCrossRefGoogle Scholar
  4. [57]
    Grothendieck, A.: Sur quelques points d’algèbre homologique. Tohoku Math. J. 9, 119–221 (1957)MATHMathSciNetGoogle Scholar
  5. [57]
    I-fausdorff, F.: Set Theory (English translation of Hausdorff [49]). New York: Chelsea 1957.Google Scholar
  6. [57]
    Kasch, F., Kneser, M., Kuppisch, H.: Unzerlegbare modulare Darstellungen endlicher Gruppen mit zyklischer p-Sylow-Gruppe. Arch. Math. 8, 320–321 (1957)MATHGoogle Scholar
  7. [57]
    Leavitt, W.: Modules without invariant basis number. Proc. Amer. Math. Soc. 7, 322–328 (1957).MathSciNetCrossRefGoogle Scholar
  8. [66]
    Balcerzyk, S.: On projective dimension of direct limits of modules. Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 14, 241–244 (1966). Bar-Hillel, Y., Fraenkel, A. A. (see Fraenkel and Bar-Hillel).Google Scholar
  9. [60]
    Bourbaki, N.: Eléments de Mathématique. Théorie des Ensembles, Chaps. 1 and 2. (A.S.I. No. 1212), Paris: Hermann 1960.Google Scholar
  10. [60]
    Chase, S. U.: Direct products of modules. Trans. Amer. Math. Soc. 97, 457–473 (1960).MathSciNetCrossRefGoogle Scholar
  11. [60]
    Eilenberg, S.: Abstract description of some basic functors. J. Indian Math. Soc. 24, 221–234 (1960).MathSciNetGoogle Scholar
  12. [60]
    Freyd, P.: Functor Theory. Ph. D. Thesis, Columbia University 1970.Google Scholar
  13. [60]
    Gentile, E.: On rings with one-sided fields of quotients. Proc. Amer. Soc. 11, 380–384 (1960).MATHMathSciNetGoogle Scholar
  14. [60]
    Goldie, A. W.: Semi-prime rings with maximum condition. Proc. Lond. Math. Soc. X, 201–220 (1960).Google Scholar
  15. [60]
    Halmos, P.: Naive Set Theory. New York: Van Nostrand 1960.Google Scholar
  16. [60]
    Lubkin, S.: Imbedding of abelian categories. Trans. Amer. Math. Soc. 97, 410–417 (1960).MathSciNetCrossRefGoogle Scholar
  17. [60]
    Northcott, D. G.: Homological Algebra. Cambridge: Cambridge University Press 1960.MATHGoogle Scholar
  18. [60]
    Von Neumann, J.: Continuous Geometry. Princeton Mathematical Series No. 25. Princeton: Princeton University 1960.Google Scholar
  19. [60]
    Watts, C.: Intrinsic characterizations of some additive functors. Proc. Amer. Math. Soc. 11, 5–8 (1960).MATHMathSciNetCrossRefGoogle Scholar
  20. [61]
    Albrecht, F.: On projective modules over semi-hereditary rings. Proc. Amer. Math. Soc. 12, 638–639 (1961).MATHMathSciNetCrossRefGoogle Scholar
  21. [61]
    Bourbaki, N.: Eléments de Mathématique. Algebra Commutative, Chaps. 1 (Modules Plat) and 2 (Localisation). (A.S.I. No. 1290), Paris: Hermann 1961.Google Scholar
  22. [61]
    Clifford, A. H., Preston, G. B. Algebraic Theory of Semigroups, Vol. I. Surveys of the Amer. Math. Soc., Vol. 7. Providence 1961.Google Scholar
  23. [61]
    Goldie, A. W.: Rings with maximum condition (Lecture Notes). Yale University Math. Dept. 1961. Bibliography 543Google Scholar
  24. [61]
    Johnson, R. E., Wong, E. T.: Quasi-injective modules and irreducible rings. J. Lond. Math. Soc. 36, 260–268 (1961).MATHMathSciNetCrossRefGoogle Scholar
  25. [71]
    Takeuchi, M.: A simple proof of Gabriel and Popesco’s theorem. J. Algebra 18 112–113 (1971).MATHMathSciNetCrossRefGoogle Scholar
  26. [71]
    Woods, S. M.: On perfect group rings. Proc. Amer. Math. Soc. 27, 49 52 (1971). [300 B.C.] Xerxes, C.: The Athenian plateau problem. Archiv. Manuskriptus 1, 1–100 (300 B.C.).Google Scholar
  27. [56]
    Cartan, H., Eilenberg, S.: Homological Algebra. Princeton: Princeton University Press 1956.MATHGoogle Scholar
  28. [56]
    Chevalley, C.: Fundamental Concepts of Algebra. New York: Academic Press 1956.MATHGoogle Scholar
  29. [56]
    Eilenberg, S.: Homological dimension and syzygies. Ann. Math. 64, 328–336 (1956).MATHMathSciNetCrossRefGoogle Scholar
  30. [56]
    Harada, M.: Note on the dimension of modules and algebras. 3. Inst. Polytechnics Osaka City University, Ser. A7, 17–27 (1956).MathSciNetGoogle Scholar
  31. [56]
    Hochschild, G.: Relative homological algebra. Trans. Amer. Math. Soc. 82, 246–269 (1956).MATHMathSciNetCrossRefGoogle Scholar
  32. [56]
    Utumi, Y.: On quotient rings. Osaka Math. J. 8, 1–18 (1956).Google Scholar
  33. [50]
    Artin, E.: The influence of J.H.M. Wedderburn on the development of modern algebra. Bull. Amer. Math. Soc. 56, 65–72 (1950).MATHMathSciNetCrossRefGoogle Scholar
  34. [50]
    Jacobson, N.: Some remarks on one-sided inverses. Proc. Amer. Math. Soc. 1, 352–355 (1950).MATHMathSciNetCrossRefGoogle Scholar
  35. [50]
    MacLane, S.: Duality in groups. Bull. Amer. Math. Soc. 56, 485–516 (1950).MathSciNetCrossRefGoogle Scholar
  36. [50]
    Spekcer, E.: Additive Gruppen von Folgen ganzer Zahlen. Portugaliae Math. 9 131–140 (1950).MathSciNetGoogle Scholar
  37. [50]
    Szele, T.: Ein Analogon der Körpertheorie für abelsche Gruppen. J.reine u. angew. Math. 188, 167–192 (1950).MATHMathSciNetCrossRefGoogle Scholar
  38. [58]
    Findlay, G. D., Lambek, J.: A generalized ring of quotients, I, II. Canad. Math. Bull. 1, 77-85, 155–167 (1958).Google Scholar
  39. [58]
    Fraenkel, A. A., Bar-Hillel, Y.: Foundations of Set Theory. Amsterdam: North Holland Publishing Comp. 1958.Google Scholar
  40. [58]
    Goldie, A. W.: The structure of prime rings under ascending chain conditions. Proc. Lond. Math. Soc. VIII, 589–608 (1958).Google Scholar
  41. [58]
    Hochschild, G.: Note on relative homological algebra. Nagoya Math. J. 13, 89-94 (1958).MATHMathSciNetGoogle Scholar
  42. [58]
    Nan, D.: Adjoint functors. Trans. Amer. Math. Soc. 87, 294–329 (1958).MathSciNetCrossRefGoogle Scholar
  43. [58]
    Lesieur, L., Croisot, R.: Théorie noethérienne des anneaux, des demi-groupes et des modules dans le cas non commutatif II. Mat. Ann. 134, 458–476 (1958).MATHMathSciNetGoogle Scholar
  44. [58]
    Morita, K.: Duality for modules and its applications to the theory of rings with minimum condition. Sci. Reports, Tokyo Kyoiku Daigaku 6, 83–142 (1958).MATHGoogle Scholar
  45. [58]
    Seshadri, C.: Trivality of vector bundles over the affine space K2. Proc. Nat. Acad. Sci. USA 44 456–458 (1958).MATHMathSciNetCrossRefGoogle Scholar
  46. [64]
    Bergman, G. M.: A ring primitive on the right but not the left. Proc. Amer. Math. Soc. 15, 473–475 (1964).MATHMathSciNetCrossRefGoogle Scholar
  47. [64]
    Brauer, R., Weiss, E.: Non-commutative Rings, Part I (Lecture Notes). Mathematics Department, Harvard University, Cambridge 1964.Google Scholar
  48. [64]
    Faith, C.: Noetherian simple rings. Bull. Amer. Math. Soc. 70, 730— 731 (1964).Google Scholar
  49. [64]
    Freyd, P.: Abelian Categories. New York: Harper Row 1964.Google Scholar
  50. [64]
    Gabriel, P., Popesco, N. (see Popesco and Gabriel).Google Scholar
  51. [64]
    Gödel, K.: The Consistency of the Axiom of Choice and the Generalized Con- tinuum Hypothesis with the Axioms of Set Theory. Ann. of Math. Studies,No. 3 (Sixth Printing). Princeton: Princeton University Press 1964.Google Scholar
  52. [64]
    Herstein, I. N.: Topics in Algebra. New York: Harper & Row 1964.Google Scholar
  53. [64]
    Jans, J. P.: Rings and Homology. New York: Holt 1964.Google Scholar
  54. [64]
    Lazard, D.: Sur les modules plats, C.R. Acad. Sci. Paris 258, 6313–6316 (1964).MATHMathSciNetGoogle Scholar
  55. [64]
    Levitzki, J.: On nil subrings. (Posthumous paper edited by S. A. Amitsur.) Israel J. Math. 1215–216 (1963).Google Scholar
  56. [64]
    Mitchell, B.: The full embedding theorem. Amer. J. Math. 86, 619–637 (1964).MATHCrossRefGoogle Scholar
  57. [64]
    Popesco, N., Gabriel, P.: Caractérisations des catégories abelienncs avec générateurs et limites inductives exactes. C. R. Acad. Sci. Paris 258, 4188–4190 (1964). Preston, G. B., Clifford, A. H. (see Clifford and Preston).Google Scholar
  58. [64]
    Sandomierski, F.: Relative injectivity and projectivity. Ph. D. Thesis, Penna, State University, University Park 1964.Google Scholar
  59. [51]
    Azumaya, G.: On maximally central algebras. Nagoya Math. J. 2, 119–150 (1951).MATHMathSciNetGoogle Scholar
  60. [51]
    Goldman, O.: Hilbert rings, and the Hilbert Nullstellensatz. Math. Z. 54, 136–140 (1951).MATHMathSciNetCrossRefGoogle Scholar
  61. [51]
    Shepherdson, J. C.: Inverse and zero divisors in matrix rings. Proc. Lond. Math. Soc. 61 71–85 (1951).MathSciNetCrossRefGoogle Scholar
  62. [69]
    Cohn, P. M.: Free associative algebras. Bull. Lond. Math. Soc. 1, 1–39 (1969).MATHCrossRefGoogle Scholar
  63. [69]
    Elizarov, V. P.: Quotient rings. Algebra and Logic 8, 219-243 (1969).MATHMathSciNetCrossRefGoogle Scholar
  64. [69]
    Goldie, A. W.: Some aspects of ring theory. Bull. Lond. Math. Soc. 1, 129–154 (1969).MATHMathSciNetCrossRefGoogle Scholar
  65. [69]
    Goldman, O.: Rings and modules of quotients. J. Alg. 13, 10–47 (1969). Goldman, O., and Auslander, M. (see Auslander and Goldman).Google Scholar
  66. [69]
    Johnson, R. E.: Extended Malcev domains. Proc. Amer. Math. Soc. 21, 211–213 (1969).MATHMathSciNetCrossRefGoogle Scholar
  67. [69]
    Lanski, C.: Nil subrings of Goldie rings are nilpotent. Canad. J. Math. 21, 904–907 (1969).MATHMathSciNetGoogle Scholar
  68. [69]
    Lenzing, H.: Endlich präsentierbare Moduln. Arch. Math. 20, 262–266 (1969).MATHMathSciNetGoogle Scholar
  69. [69]
    Mewborn, A. C., Winton, C. N.: Orders in self-injective semi-perfect rings. J. Algebra 13, 5–9 (1969).MATHMathSciNetCrossRefGoogle Scholar
  70. [69]
    Osofsky, B. L.: A commutative local ring with finite divisors. Trans. Amer. Math. Soc. 141, 377-385 (1969).MATHMathSciNetCrossRefGoogle Scholar
  71. [69]
    Vasconcelos, W. V.: On finitely generated flat modules. Trans. Amer. Math. Soc. 138, 505— 512 (1969).Google Scholar
  72. [70a]
    Vasconcelos, W. V.: Flat modules over commutative noetherian rings. Trans. Amer. Math. Soc. 152, 137–143 (1970).MATHMathSciNetCrossRefGoogle Scholar
  73. [69]
    Zelmanowitz, J.: A shorter proof of Goldie’s theorem. Canad. Math. Bull. 12, 597–602 (1969).MATHMathSciNetCrossRefGoogle Scholar
  74. [63]
    Bourbaki, N.: Eléments de Mathématique. Théorie des Ensembles, Chap. 3. (A.S.I. No. 1243) Paris: Hermann 1963.Google Scholar
  75. [63]
    Brauer, R.: Representations of finite groups, Lectures on Modern Mathematics, Vol. I (T. L. Saaty, Ed.). New York: Wiley 1953, pp. 133–175.Google Scholar
  76. [63]
    Connell, I.: On the group ring. Canad. J. Math. 15, 650–685 (1963).MATHMathSciNetGoogle Scholar
  77. [63]
    Corner, A. L. S.: Every countable reduced torsion-free ring is an endomorphism ring. Proc. Lond. Math. Soc. (3) 13, 687–710 (1963).MATHMathSciNetCrossRefGoogle Scholar
  78. [63]
    Feit, W., Thompson, J. G.: Solvability of groups of odd order. Pac. J. Math. 13, 775–1029 (1963).MATHMathSciNetGoogle Scholar
  79. [63]
    Kurosch, A. G.: General Algebra. New York: Chelsea 1963.Google Scholar
  80. [63]
    Lambek, J.: On L’tumi’s ring of quotients. Canad. J. Math. 15, 363–370 (1963).MATHMathSciNetGoogle Scholar
  81. [63]
    MacLane, S.: Homology. Berlin/Göttingen/Heidelberg: Springer 1963.MATHGoogle Scholar
  82. [63]
    Procesi, C.: On a theorem of Goldie concerning the structure of prime rings with maximal condition (in Italian). Acad. Naz. Lincei Rend. 34, 372–377 (1963).MATHMathSciNetGoogle Scholar
  83. [63]
    Solovay, R.: Independence results in the theory of cardinals (Preliminary Report). Notices of the Amer. Math. Soc. 10, 595 (1963).Google Scholar
  84. [63]
    Talintyre, T. D.: Quotient rings of rings with maximal condition for right ideals. J. Lond. Math. Soc. 38, 439–450 (1963).MATHMathSciNetCrossRefGoogle Scholar
  85. [65]
    Abian, A.: The Theory of Sets and Transfinite Arithmetic. Philadelphia: Saunders 1965.Google Scholar
  86. [65]
    Bumby, R. T.: Modules which are isomorphic to submodules of each other. Arch. math. 16, 184–185 (1965).MATHMathSciNetCrossRefGoogle Scholar
  87. [65]
    Chase, S. U., Faith, C.: Quotient rings and direct products of full linear rings. Math. Zeit. 88, 250–264 (1965).MATHMathSciNetCrossRefGoogle Scholar
  88. [65]
    Cohn, P. M.: Universal Algebra. New York: Harper Row 1965.Google Scholar
  89. [65]
    Courter, R. C.: The dimension of a maximal commutative subalgebra of K n. Duke Math. J. 32, 225–232 (1965).MATHMathSciNetGoogle Scholar
  90. [65]
    Faith, C.: Orders in simple artinian rings. Trans. Amer. Math. Soc. 114, 61–64 (1965).MATHMathSciNetCrossRefGoogle Scholar
  91. [65]
    Grothendieck, A.: Le groupe dc Brauer I. Séminaire Bourbaki, exposé 290, Paris: Hermann 1965.Google Scholar
  92. [65]
    Herstein, I. N.: A counter example in Noetherian rings. Proc. Nat. Acad. Sci. (U.S.A.) 54, 1036–1037 (1965)MATHMathSciNetCrossRefGoogle Scholar
  93. [65]
    Lang, S.: Algebra. Cambridge: Addison-Wesley 1965.MATHGoogle Scholar
  94. [65]
    Mitchell, B.: Theory of Categories. New York: Academic Press 1965.MATHGoogle Scholar
  95. [65]
    Procesi, C.: On a theorem of Faith and Utumi (in Italian). Rend. Math. e. Appl. 24, 346–347 (1965).MATHMathSciNetGoogle Scholar
  96. [65]
    Procesi, C., Small, L.: On a theorem of Goldie. J. Algebra 2, 80–84 (1965).MATHMathSciNetCrossRefGoogle Scholar
  97. [65]
    Richman, F.: Generalized quotient rings. Proc. Amer. Math. Soc. 16, 794— 799 (1965).Google Scholar
  98. [65]
    Roos, J. E.: Caractérisation des catégories qui sont quotients des catégories des modules par des sous-catégories bilocalisantes. C. R. Acad. Sci. Paris 261, 4954-4957 (1965).MATHMathSciNetGoogle Scholar
  99. [65]
    Rotman, J.: The Theory of Groups. Boston: Allyn & Bacon 1965. Rutter, E. A., Colby, R. R. (see Colby and Rutter). Saha, F., Gupta, R. N. (see Gupta and Saha).Google Scholar
  100. [65]
    Small, L. W.: An example in Noetherian rings. Proc. Nat. Sci. USA 54 1035–1036 (1965).MATHMathSciNetCrossRefGoogle Scholar
  101. [65]
    Willard, E.: Properties of generators. Math. Ann. 158, 352–364 (1965). Winton, C. N., Mewborn, A. C. (sec Mewborn and Winton). Wong, E. T., Johnson, R. E. (see Johnson and Wong).Google Scholar
  102. [71]
    Boyle, A. K.: Ph. D. Thesis, 1971. Rutgers University, New Brunswick.Google Scholar
  103. [71]
    Cohn, P. M.: Free Rings and Their Relations. New York: Academic Press 1971.MATHGoogle Scholar
  104. [71]
    Colby, R. R., Rutter, E. A., Jr.: 17-Flat and 17-Projective modules. Arch. Math. 22, 246–251 (1971).MATHMathSciNetGoogle Scholar
  105. [71]
    Dade, E. C.: Deux groupes finis distincts ayant la même algebre de groupe sur tout corps. Math. Z. 119, 345–348 (1971).MATHMathSciNetCrossRefGoogle Scholar
  106. [71]
    Eisenbud, D., Griffith, P.: Serial rings. J. Algebra 17, 389–400 (1971).MATHMathSciNetCrossRefGoogle Scholar
  107. [71]
    Fuller, K. R.: Primary rings and double centralizers. Pac. J. Math. 34, 379-383 (1970). Fuller, K. R., Camillo, V. (see Camillo and Fuller).Fuller, K. R., Dickson, S. E. (see Dickson and Fuller).Google Scholar
  108. [71]
    Lambek, J.: Torsion Theories, Additive Semantics, and Rings of Quotients. Lecture Notes in Mathematics, No. 177. Berlin/Heidelberg/New York: Springer 1971. Lambek, J., Findlay, G. D. (see Findlay and Lambek).Google Scholar
  109. [71]
    MacLane, S.: Categorical Algebra and set-theoretical foundations. Amer. Math. Soc. Proceedings of Symposia in Pure Math. Vol. XIII, Part I, 231–240 (1971).MathSciNetGoogle Scholar
  110. [71]
    Scott, D. (editor): Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics, Vol. 13, Part I. Amer. Math. Soc., Providence 1971.Google Scholar
  111. [71]
    Stenstrom, B.: Rings and modules of quotients. Lecture Notes in Mathematics, No. 237. Berlin/Heidelberg/New York: Springer 1971Google Scholar
  112. [71]
    Takeuchi, M.: A simple proof of Gabriel and Popesco’s theorem. J. Algebra 18 112–113 (1971).MATHMathSciNetCrossRefGoogle Scholar
  113. [71]
    Woods, S. M.: On perfect group rings. Proc. Amer. Math. Soc. 27, 49 52 (1971). [300 B.C.] Xerxes, C.: The Athenian plateau problem. Archiv. Manuskriptus 1, 1–100 (300 B.C.).Google Scholar
  114. [69a]
    Jategaonkar, A. V.: Ore domains and free algebras. Bull. Lond. Math. Soc. 1, 45-46 (1969).MATHMathSciNetGoogle Scholar
  115. [69a]
    Kaplansky, I.: Fields and Rings, Chicago Lectures in Mathematics. Chicago/ London: University of Chicago Press 1969.Google Scholar
  116. [69a]
    Michler, G.: On quasi-local noetherian rings. Proc. Amer. Math. Soc. 20, 222–224 (1969).MATHMathSciNetCrossRefGoogle Scholar
  117. [44]
    Artin, E., Nesbitt, E., Thrall, R.: Rings with Minimum Condition. Ann Arbor: University of Michigan Press 1944.MATHGoogle Scholar
  118. [44]
    Weyl, H.: David Hilbert and his mathematical work. Bull. Amer. Math. Soc. 50, 612–654 (1944). Whaples, G. (sec Artin and Whaples).Google Scholar
  119. [44]
    Whitney, H.: Topics in the theory of abelian groups. I. Divisibility of homomorphisms. Bull. Amer. Math. Soc. 50, 129–144 (1944).MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag, Berlin · Heidelberg 1973

Authors and Affiliations

  • Carl Faith
    • 1
  1. 1.Rutgers UniversityNew BrunswickUSA

Personalised recommendations