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Torsion-Free and Divisible Modules Over Non-Integral-Domains

Published online by Cambridge University Press:  20 November 2018

Lawrence Levy
Lawrence Levy*
Affiliation:
University of Illinois and University of Wisconsin
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In trying to extend the concept of torsion to rings more general than commutative integral domains the first thing that we notice is that if the definition is carried over word for word, integral domains are the only rings with torsion-free modules. Thus, if m is an element of any right module M over a ring containing a pair of non-zero elements x and y such that xy = 0, then either mx = 0 or (mx)y = 0. A second difficulty arises in the non-commutative case: Does the set of torsion elements of M form a submodule? The answer to this question will not even be "yes" for arbitrary non-commutative integral domains.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 15 , 1963 , pp. 132 - 151
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Artin, E., Nesbitt, C., and Thrall, R., Rings with minimum condition (Ann Arbor, 1944).Google Scholar
2. Cartan, H. and Eilenberg, S., Homological algebra (Princeton, 1956).Google Scholar
3. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (New York, 1962).Google Scholar
4. Gentile, E., On rings with one-sided field of quotients, Proc. Amer. Math. Soc, 11 (1960), 380–384.Google Scholar
5. Goldie, A. W., The structure of prime rings under ascending chain conditions, Proc. London Math. Soc, Third Ser., 8, no. 32 (1958), 589–608.Google Scholar
6. Goldie, A. W., Semi-prime rings with maximum condition, Proc. Amer. Math. Soc, Third Ser., 10, no. 38 (1960), 201–220.Google Scholar
7. Jacobson, N., The theory of rings (New York, 1943).Google Scholar
8. Jacobson, N., Structure of rings (Providence, 1956).Google Scholar
9. Kaplansky, I., Modules over Dedekind and valuation rings, Trans. Amer. Math. Soc, 72 (1952), 327–340.Google Scholar
10. Levy, L. S., Unique subdirect sums of prime rings, to appear in Trans. Amer. Math. Soc.Google Scholar
11. Matlis, E., Infective modules over noetherian rings, Pac J. Math., 8 (1958), 511–528.Google Scholar