Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-27T22:29:43.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Notes on weakly-semisimple rings

Published online by Cambridge University Press:  17 April 2009

Yiqiang Zhou
Yiqiang Zhou
Affiliation:
Department of MathematicsUniversity of British ColumbiaVancouver BCV6T 1Z2Canada.
Rights & Permissions [Opens in a new window]

Abstract

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.

Responding to a question on right weakly semisimple rings due to Jain, Lopez-Permouth and Singh, we report the existence of a non-right-Noetherian ring R for which every uniform cyclic right it-module is weakly-injective and every uniform finitely generated right R-module is compressible. We show that a ring R is a right Noetherian ring for which every cyclic right R-module is weakly R-injective if and only if R is a right Noetherian ring for which every uniform cyclic right R-module is compressible if and only if every cyclic right R-module is compressible. Finally, we characterise those modules M for which every finitely generated (respectively, cyclic) module in σ[M] is compressible.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 52 , Issue 3 , December 1995 , pp. 517 - 525
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Al-Huzali, A., Jain, S.K. and Lopez-Permouth, S.R., ‘On the weak relative-injectivity of rings and modules’, in Noncommutative ring theory, Lecture Notes in Math. 1448 (Springer-Verlag, Berlin, Heidelberg, New York, 1990), pp. 93–98.CrossRefGoogle Scholar
[2]Boyle, A.K., ‘Injectives containing no proper quasi-injective submodules’, Comm. Algebra 4 (1976), 775–785.CrossRefGoogle Scholar
[3]Golan, J.S. and Lopez, S.R.-Permouth, ‘QI-filters and tight modules’, Comm. Algebra 19 (1991), 2217–2229.CrossRefGoogle Scholar
[4]Goodearl, K.R. and Warfield, R.B. Jr, An introduction to noncommutative Noetherian rings (London Math. Soc, Cambridge Univ. Press, Cambridge, 1989).Google Scholar
[5]Jain, S.K. and Lopez-Permouth, S.R., ‘Rings whose cyclics are essentially embeddable in projective modules’, J. Algebra 128 (1990), 257–269.CrossRefGoogle Scholar
[6]Jain, S.K. and Lopez-Permouth, S.R., ‘A survey on the theory of weakly-injective modules’, in Computational Algebra, Lecture Notes in Pure and Appl. Math. 151 (Marcel Dekker, New York, 1994), pp. 205–232.Google Scholar
[7]Jain, S.K., Lopez-Permouth, S.R. and Singh, S., ‘On a class of QI-rings’, Glasgow Math. J. 34 (1992), 75–81.CrossRefGoogle Scholar
[8]Jain, S.K. and Mohamed, S., ‘Rings whose cyclic modules are continuous’, J. Indian Math. Soc. 42 (1978), 197–202.Google Scholar
[9]Jategaonkar, A.V., Localization in Noetherian rings (Cambridge University Press, Cambridge, London, New York, 1986).CrossRefGoogle Scholar
[10]Levy, L., ‘Torsion-free and divisible modules over non-integral-domains’, Canad. J. Math. 15 (1963), 132–151.CrossRefGoogle Scholar
[11]Lopez-Permouth, S.R., ‘Rings characterized by their weakly-injective modules’, Glasgow Math. J. 34 (1992), 349–353.CrossRefGoogle Scholar
[12]Lopez-Permouth, S.R., Rizvi, S.T. and Yousif, M.F., ‘Some characterizations of semiprime Goldie rings’, Glasgow Math. J. 35 (1993), 357–365.CrossRefGoogle Scholar
[13]Shock, R.C., ‘Dual generalizations of the Artinian and Noetherian conditions’, Pacific J. Math. 54 (1974), 227–235.CrossRefGoogle Scholar
[14]Wisbauer, R., ‘Generalized co-semisimple modules’, Comm. Algebra 8 (1990), 4235–4253.CrossRefGoogle Scholar
[15]Zhou, Y., ‘Strongly compressible modules and semiprime right Goldie rings’, Comm. Algebra 21 (1993), 687–698.CrossRefGoogle Scholar