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On semi-artinian modules and injectivity conditions

Published online by Cambridge University Press:  20 January 2009

J. Clark  and
P. F. Smith
J. Clark
Affiliation:
Department of Mathematics and Statistics University of Otago P. O. Box 56 Dunedin New Zealand E-mail: jclark@maths.otago.ac.nz
P. F. Smith
Affiliation:
Department of Mathematics University of Glasgow University Gardens Glasgow G12 8QW Scotland E-mail: pfs@maths.gla.ac.uk
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Abstract

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It is well known that a module M has finite length if and only if it is semi-artinian and Noetherian or, equivalently, semi-noetherian and artinian. Our main result shows that finite length is often achieved by just assuming that M is semi-artinian, semi-noetherian and has finitely generated socle.

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 2 , June 1996 , pp. 263 - 270
Copyright
Copyright © Edinburgh Mathematical Society 1996

References

REFERENCES

1. Anderson, F. W. and Fuller, K. R. Rings and categories of modules (Springer-Verlag, Berlin, 1974).CrossRefGoogle Scholar
2. Ánh, Phạm Ngọc, Morita duality for commutative rings, Comm. Algebra 18 (1990), 1781–1788.CrossRefGoogle Scholar
3. Armendariz, E. P., Rings with DCC on essential left ideals, Comm. Algebra 8 (1980), 299–308.CrossRefGoogle Scholar
4. Baccella, G., Semi-artinian V-rings and semi-artinian von Neumann regular rings, J. Algebra, to appear.Google Scholar
5. Bass, H., Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.CrossRefGoogle Scholar
6. Camillo, V. P. and Fuller, K. R., On Loewy length of rings, Pacific J. Math. 53 (1974), 347–354.CrossRefGoogle Scholar
7. Chatters, A. W. and Hajarnavis, C. R. Rings with chain conditions (Pitman, London, 1980).Google Scholar
8. Clark, J., On a question of Faith in commutative endomorphism rings, Proc. Amer. Math. Soc. 98 (1986), 196–198.CrossRefGoogle Scholar
9. Clark, J. and Van Huynh, Dinh, A note on perfect self-injective rings, Quart. J. Math. Oxford 45 (1994), 13–17.CrossRefGoogle Scholar
10. Faith, C., Algebra II: Ring Theory (Springer-Verlag, Berlin, 1976).CrossRefGoogle Scholar
11. Faith, C., Lineary compact injective modules and a theorem of Vámos, Publ. Sec. Math. Univ. Autònoma Barcelona 30 (1986), 127–148.Google Scholar
12. Faith, C., Polynomial rings over Jacobson-Hilbert rings, Publ. Sec. Math. Univ. Autònoma Barcelona 33 (1989), 85–97.Google Scholar
13. Fossum, R. M., Griffith, P. A. and Reiten, I., Trivial Extensions of Abelian Categories (Lecture Notes in Mathematics 456, Springer-Verlag, Berlin, 1975).CrossRefGoogle Scholar
14. Hernández, J. L. Garcīa and Pardo, J. L. Gómez, On endomorphism rings of quasiprojec-tive modules, Math. Z. 196 (1987), 87–108.Google Scholar
15. Gupta, A. K. and Varadarajan, K., Modules over endomorphism rings, Comm. Algebra 8 (1980), 1291–1333.CrossRefGoogle Scholar
16. Jans, J. P., On co-Noetherian rings, J. London Math. Soc. (2) 1 (1969), 588–590.CrossRefGoogle Scholar
17. Jategaonkar, A. V., Certain injectives are Artinian, in Noncommutative ring theory (Intern. Conf., Kent State Univ., Kent, Ohio, 1975) (Lecture Notes in Mathematics 545, Springer-Verlag, Berlin, 1976), 128–139.CrossRefGoogle Scholar
18. Matlis, E., Injective modules over Prüfer rings, Nagoya Math. J. 15 (1959), 57–69.CrossRefGoogle Scholar
19. Menini, C. and Orsatti, A., Topologically left Artinian rings, J. Algebra 93 (1985), 475–508.CrossRefGoogle Scholar
20. Michler, G. O. and Villamayor, O. E., On rings whose simple modules are infective, J. Algebra 25 (1973), 185–201.CrossRefGoogle Scholar
21. Müller, B. J., Linear compactness and Morita duality, J. Algebra 16 (1970), 60–66.CrossRefGoogle Scholar
22. Musson, I. M., Injective modules for group rings of polycyclic groups I, Quart. J. Math. Oxford 31 (1980), 429–448.CrossRefGoogle Scholar
23. Musson, I. M., Injective modules for group rings of polycyclic groups II, Quart. J. Math. Oxford 31 (1980), 449–466.CrossRefGoogle Scholar
24. Năstăsescu, C. and Popescu, N., Anneaux semi-artiniens, Bull. Soc. Math. France 96 (1968), 357–368.CrossRefGoogle Scholar
25. Singh, S., Quasi-injective and quasi-projective modules over hereditary Noetherian prime rings, Canad. J. Math. 26 (1974), 1173–1185.CrossRefGoogle Scholar
26. Vámos, P., The dual of the notion of “finitely generated”, J. London Math. Soc. 43 (1968), 643–646.CrossRefGoogle Scholar
27. Vámos, P., Classical rings, J. Algebra 34 (1975), 114–129.CrossRefGoogle Scholar
28. Weimin, Xue, Rings with Morita Duality (Lecture Notes in Mathematics 1523, Springer-Verlag, Berlin, 1992).Google Scholar