Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-15T06:53:52.230Z Has data issue: false hasContentIssue false
  • English
  • Français

Modules Over Hereditary Noetherian Prime Rings

Published online by Cambridge University Press:  20 November 2018

Surjeet Singh
Surjeet Singh*
Affiliation:
Ohio University, Athens, Ohio; Guru Nanak University, Amritsar, 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.

Quasi-injective and quasi-projective modules over hereditary noetherian prime rings ((hnp)-rings) were studied in [17]. In the present paper we give some applications of the results established in [17]. Kulikov, Kertesz, Prufer, Szele had made basic contributions to the problem of decomposability of abelian p-groups (Fuchs [4]). Kaplansky [9] studied analogous problems for modules over (commutative) Dedekind domains. Let R be an (hnp)-r'mg, which is not right primitive. Using the structure of an indecomposable infective torsion R-module, established in [17, Theorem 4], some of the basic concepts and results on the decomposability of a torsion abelian group are generalized in Section 2, to modules over R.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 4 , 01 August 1975 , pp. 867 - 883
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Eisenbud, D. and Robson, J. C., Modules over Dedekind prime rings, J. Algebra 16 (1970), 67–85.Google Scholar
2. Eisenbud, D. and Robson, J. C., Hereditary noetherian prime rings, J. Algebra 16 (1970), 86–104.Google Scholar
3. Eisenbud, D. and Griffith, P., Serial rings, J. Algebra 17 (1971), 389–400.Google Scholar
4. Fuchs, L., Abelian groups (Pergamon Press, New York, 1960).Google Scholar
5. Fuchs, L. and Rangaswamy, K. M., Quasi-projective abelian groups, Bull. Soc. Math. France 98 (1970), 5–8.Google Scholar
6. Fuller, K. R. and Hill, D. A., Quasi-projective modules via relative projectivity, Arch. Math. 21 (1970), 369–373.Google Scholar
7. Goldie, A. W., Semi prime rings with maximum condition, Proc. London Math. Soc. 10 (1960), 201–220.Google Scholar
8. Jataegaonker, A. V., Left principal ideal rings, Lecture Notes in Mathematics No. 123 (Springer Verlag, 1970).Google Scholar
9. Kaplansky, I., Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72 (1952), 327–340.Google Scholar
10. Lenagan, T. H., Bounded hereditary noetherian prime rings, J. London Math. Soc. 6 (1973), 241–246.Google Scholar
11. Levy, L. S., Torsion free and divisible modules over non-integral domains, Can. J. Math. 15 (1963), 132–151.Google Scholar
12. Marubayashi, H., Modules over bounded Dedekind prime rings, Osaka J. Math. 9 (1972), 95–110.Google Scholar
13. Marubayashi, H., Modules over bounded Dedekind prime rings, II, Okaka J. Math. 9 (1972), 427–445.Google Scholar
14. Matlis, E., Injective modules over noetherian rings, Pacific J. Math. 8 (1958), 511–528.Google Scholar
15. Rangaswamy, K. M. and Vanaja, N., Quasi-projectives in abelian categories, Pacific J. Math. 43 (1972), 221–238.Google Scholar
16. deRobert, E., Projectifs et injeitifs relatifs, C. R. Acad. Sci. Paris Sér. A-B, Vol. 286 (1969), 361–364.Google Scholar
17. Singh, S., Quasi-injective and quasi-projective modules over hereditary noetherian prime rings, Can. J. Math. 26 (1974), 1173–1185.Google Scholar
18. Small, L. W., Semi-hereditary rings, Bull. Amer. Math. Soc. 73 (1967), 656–658.Google Scholar