Skip to main content Accessibility help

Definability problems for modules and rings1

  • Gabriel Sabbagh (a1) and Paul Eklof (a1)


This paper is concerned with questions of the following kind: let L be a language of the form Lαω and let be a class of modules over a fixed ring or a class of rings; is it possible to define in L? We will be mainly interested in the cases where L is Lωω or L∞ω and is a familiar class in homologic algebra or ring theory.

In Part I we characterize the rings Λ such that the class of free (respectively projective, respectively flat) left Λ-modules is elementary. In [12] we solved the corresponding problems for injective modules; here we show that the class of injective Λ-modules is definable in L∞ω if and only if it is elementary. Moreover we identify the right noetherian rings Λ such that the class of projective (respectively free) left Λ-modules is definable in L∞ω.



Hide All

During the writing of this paper at Yale University the second author was supported by a grant of the Lebanese Council for Scientific Research.



Hide All
[1]Azumaya, G., Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt's theorem, Nagoya Mathematical Journal, vol. 1 (1950), pp. 117124.
[2]Barwise, J. and Eklof, P., Infinitary properties of abelian torsion groups. Annals of Mathematical Logic vol. 2 (1970), pp. 2568.
[3]Bass, H., Finitistic dimension and a homological generalization of semiprimary rings, Transactions of the American Mathematical Society, vol. 95 (1960), pp. 466488.
[4]Björk, J. E., Rings satisfying a minimum condition on principal ideals, Journal für die reine und angewandte Mathematik, vol. 236 (1969), pp. 112119.
[4]Bourbaki, N., Algèbre, Chap. 2, Algèbre linéaire, troisième edition, Hermann, Paris, 1962.
[5]Bourbaki, N., Algèbre, Chap. 8, Modules et anneaux semi-simples, Hermann, Paris, 1958.
[6]Bourbaki, N., Algèbre commutative, Chaps. 1 and 2, Hermann, Paris, 1961.
[7]Cartan, H. and Eilenberg, S., Homological algebra, Princeton University Press, Princeton, N.J., 1956.
[8]Chase, S. U., Direct products of modules, Transactions of the American Mathematical Society, vol. 97 (1960), pp. 457473.
[9]Cohn, P. M., On the free product of associative rings, Mathematische Zeitschrift, vol. 71 (1959), pp. 380398.
[10]Cohn, P. M., Some remarks on the invariant basis property, Topology, vol. 5 (1966), pp. 215228.
[11]Eilenberg, S., Homological dimension and syzygies, Annals of Mathematics, vol. 64 (1956), pp. 328336.
[12]Eklof, P. and Sabbagh, G., Model-completions and modules, Annals of Mathematical Logic, vol. 2 (1971), pp. 251295.
[13]Feferman, S. and Vaught, R. L., The first order properties of products of algebraic systems, Fundamenta Mathematicae, vol. 47 (1959), pp. 57103.
[14]Fuchs, L., Abelian groups, Pergamon Press, Oxford, 1960.
[15]Govorov, V. E., Rings over which flat modules are free (Russian), Doklady Akademii Nauk SSSR, vol. 144 (1962), pp. 965967.
[16]Jategaonkar, A. V., A counter-example in ring theory and homological algebra, Journal of Algebra, vol. 12 (1969), pp. 418440.
[17]Kaplansky, I., Projective modules, Annals of Mathematics, vol. 68 (1958), pp. 372377.
[18]Karp, C. R., Languages with expressions of infinite length, North-Holland, Amsterdam, 1964.
[19]Karp, C. R., Finite-quantifier equivalence, The theory of models, North-Holland, Amsterdam, 1965.
[20]Lenzing, H., Direkte Produkte von freien Moduln, Mathematische Zeitschrift, vol. 106 (1968), pp. 206212.
[21]Lenzing, H., Über kohärente Ringe, Mathematische Zeitschrift, vol. 114 (1970), pp. 201212.
[22]Lopez-Escobar, E. G. K., An addition to: “On defining well-orderings”, Fundamenta Mathematicae, vol. 59 (1966), pp. 299300.
[23]Matlis, E., Injective modules over Noetherian rings, Pacific Journal of Mathematics, vol. 8 (1958), pp. 511528.
[24]Morita, K., Kawada, Y. and Tachikawa, H., On injective modules, Mathematische Zeitschrift, vol. 68 (1957), pp. 217226.
[25]Osofsky, B. L., A generalization of quasi-Frobenius rings, Journal of Algebra, vol. 4 (1966), pp. 373387.
[26]Szmtelew, W., Elementary properties of Abelian groups, Fundamenta Mathematicae, vol. 41 (1955), pp. 203271.
[27]Zariski, O. and Samuel, P., Commutative Algebra, vol. I, Van Nostrand, Princeton, N.J., 1958.

Definability problems for modules and rings1

  • Gabriel Sabbagh (a1) and Paul Eklof (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed