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A Generalization of Divisibility and Injectivity in Modules

Published online by Cambridge University Press:  20 November 2018

D. F. Sanderson
D. F. Sanderson*
Affiliation:
Western Washington State College, BellinghamWashington
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Classically, there has been, for obvious reasons, an intimate relation between the concepts "rings of quotients" and "divisible modules". Recently, however, their generalizations have appeared to diverge.

For example, Hattori ([9]) and Levy ([15]) have generalized the concept of "divisibility" as follows: Hattori (respectively Levy) defines a left R-module M over a ring R to be divisible if and only if Ext1R(R/I, M)=0 for every principal left ideal I ⊂ R (respectively, every principal left ideal I ⊂ R which is generated by a regular element of R).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 8 , Issue 4 , June 1965 , pp. 505 - 513
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Baer, R., Abulias subgroups which are direct summands of every containing abelian group, Bull. Am. Math. Soc., 46 (1940), 800-806.Google Scholar
2. Bourbaki, N. Éléments de mathématique, Vol.27 (Paris 1961), 157-165.Google Scholar
3. Butler, M. C. R. and Horrocks, G., Classes of extensions and resolutions, Phil. Trans. Royal Soc. London, 254 (1961), 155-222.Google Scholar
4. Cartan, H. and Eilenberg, S., Homological Algebra, (Princeton 1956).Google Scholar
5. Eckmann, B. and Schopf, A., Űber injektive moduln, Arch. Math., 4(1965), 75-78.Google Scholar
6. Faith, C. and Utumi, Y., Baer modules, Arch. Math., 15 (1964), 266-270.Google Scholar
7. Goldie, A.W., Rings with maximum condition, Yale University lecture notes (1962).Google Scholar
8. Goldie, A. W., Semi-prime rings with maximum condition, Proc. London Math. Soc., 10(1960), 201-220.Google Scholar
9. Hattori, A., A foundation of torsion theory for modules over general rings, Nagoya Math. J., 17 (1960), 147-158.Google Scholar
10. Herstein, I.N. and Small, L., Nil rings satisfying certain chain conditions, Can. J. Math., 16(1964), 771-776.Google Scholar
11. Jacobson, N., The theory of rings, (New York, 1943).Google Scholar
12. Johnson, R. E., The extended centralizer of a ring over a module, Proc. Amer. Math. Soc., 2(1951), 891-895.Google Scholar
13. Kertész, A., Systems of equations over modules, Acta Sci. Math. Szeged, 18 (1957), 207-234.Google Scholar
14. Lambek, J., On Utumi' s ring of quotients, Can. J. Math., 15 (1963), 363-370.Google Scholar
15. Levy, L., Torsion-free and divisible modules over nonintegral domains, Can. J. Math., 15(1963), 132-151.Google Scholar
16. Utumi, Y., On quotient rings, Osaka Math. J., 8 (1956), 1-18.Google Scholar
17. Gentile, E. R., Singular submodule and injective hull, Indag. Math., 24 (1962), 426-433.Google Scholar
18. Maranda, J. M., Injective structures, Trans. Am. Math. Soc., 110 (1964), 98-135.Google Scholar