Hostname: page-component-7479d7b7d-jwnkl Total loading time: 0 Render date: 2024-07-11T16:18:54.976Z Has data issue: false hasContentIssue false
  • English
  • Français

Co-absolutely co-pure modules

Published online by Cambridge University Press:  20 January 2009

V. A. Hiremath
V. A. Hiremath
Affiliation:
Department of Mathematics, Madurai Kamaraj University, Madurai-625 021, India
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

B. Maddox [15] defined absolutely pure modules and derived some interesting properties of these modules. C. Megibben [17] continued the study of these modules and found more interesting properties. We introduce in this paper co-absolutely co-pure modules as dual to absolutely pure modules. We first prove that over a commutative classical ring these modules are precisely the flat modules. As a biproduct we get a projective characterization of flat modules over a commutative co-noetherian ring. Secondly, over a quasi-Frobenius ring R, co-absolutely co-pure right R-modules turn out to be projective modules. Finally we get a characterization of almost Dedekind domains in terms of co-absolutely co-pure modules.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 29 , Issue 3 , October 1986 , pp. 289 - 298
Copyright
Copyright © Edinburgh Mathematical Society 1986

References

REFERENCES

1.Bass, H., Finitistic dimension and homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960, 466–488.CrossRefGoogle Scholar
2.Cartan, H. and Eilenberg, S., Homological Algebra (Princeton University Press, 1956).Google Scholar
3.Chase, S. U., Direct product of modules, Trans. Amer. Math. Soc. 97 (1960), 457–473.CrossRefGoogle Scholar
4.Cohn, P. M., On the free product of associative rings, Math. Z. 71 (1959), 380–398.CrossRefGoogle Scholar
5.Cozzens, J. H., Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75–79.CrossRefGoogle Scholar
6.Endo, S., On semi-hereditary rings, J. Math. Soc. Japan 13 (1961), 109–119.CrossRefGoogle Scholar
7.Faith, C., Albegra I: Rings, Modules and Categories (Springer-Verlag, Berlin, Heidelberg, New York, 1973).Google Scholar
8.Faith, C., Algebra II: Ring Theory (Springer-Verlag, Berlin, Heidelberg, New York, 1976).CrossRefGoogle Scholar
9.Gilmer, R., Multiplicative Ideal Theory (Marcel-Dekker Inc., New York, 1971).Google Scholar
10.Hiremath, V. A., Cofinitely generated and cofinitely related modules, Acta Math. Hungar. 39 (1982), 1–9.CrossRefGoogle Scholar
11.Hiremath, V. A., Copure submodules, Acta Math. Hungar. 44 (1984), 3–12.CrossRefGoogle Scholar
12.Hiremath, V. A., Confinitely projective modules, Houston J. Math. 11 (1985), 183–190.Google Scholar
13.Hiremath, V. A., Copure-injective modules, Glasgow Math. J., submitted for publication.Google Scholar
14.Jans, J. P., On co-noetherian rings, J. London Math. Soc. (2) 1 (1969), 588–590.CrossRefGoogle Scholar
15.Maddox, B., Absolutely pure modules, Proc. Amer. Math. Soc. 18 (1967), 155–158.CrossRefGoogle Scholar
16.Mccoy, N. H., Theory of Rings (Macmillan, New York, 1964).Google Scholar
17.Megibben, C., Absolutely pure modules, Proc. Amer. Math. Soc. 26 (1970), 561–566.CrossRefGoogle Scholar
18.Mitchell, B., Theory of Categories (Academic Press, New York, 1964).Google Scholar
19.Osofosky, B. L., Cyclic injective modules of full linear rings, Proc. Amer. Math. Soc. 17 (1966), 247–253.CrossRefGoogle Scholar
20.Rotman, J. J., Notes on Homological Algebra (Van Nostrand Reinhold Company, New York, 1968).Google Scholar
21.Stestrom, B., Rings of Quotients (Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
22.Vamos, P., On the dual of the notion of “finitely generated”, L. London Math. Soc. 43 (1968), 643–646.CrossRefGoogle Scholar
23.Vamos, P., Classical rings, J. Algebra 34 (1975), 114–129.CrossRefGoogle Scholar
24.Warfield, R. B. Jr., Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), 699–719.CrossRefGoogle Scholar