Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T05:58:13.391Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak q-Rings

Published online by Cambridge University Press:  20 November 2018

Saad Mohamed  and
Surjeet Singh
Saad Mohamed
Affiliation:
Kuwait University, Kuwait
Surjeet Singh
Affiliation:
Guru Nanak Dev 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.

Throughout this paper we assume that every ring has unity and all modules are unital right modules. A ring R is called a (right) q-ring if every right ideal of R is quasi-injective [4], In this paper we study a generalization of this concept. A ring R is called a (right) weak q-ring (in short, wq-ring) if every right ideal of R, not isomorphic to RR, is quasi-injective. A ring R is called a right pq-ring if every proper right ideal of R is quasi-injective. Any upper triangular 2 X 2 matrix ring over a division ring is a wq-ring, which is not a q-ring. In Section 1, some general properties of wq-rings are established and, in particular, it is shown in (1.8) that a semiprime wq-ring has zero singular ideal.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 4 , 01 August 1977 , pp. 687 - 695
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Bumby, R. T., Modules which are isomorphic to submodules of each other, Arch. Math. 16 (1965), 184–185.Google Scholar
2. Goldie, A. W., Semiprime rings with maximum conditions, Proc. London, Math. Soc. 10 (1960), 201–220.Google Scholar
3. Harada, M., Note on quasi-injective modules, Osaka Math J. 2 (1965), 351–356.Google Scholar
4. Jain, S. K., S. H. Mohamed and Surjeet Singh, Rings in which every right ideal is quasiinjective, Pacific J. Math. 31 (1969), 73–79.Google Scholar
5. Johnson, R. E. and Wong, E. T., Self-injective rings, Canadian Math. Bull. 2 (1959), 167–173.Google Scholar
6. Johnson, R. E. and Wong, E. T. Quasi-injective modules and irreducible rings, J. London, Math. Soc. 36 (1961), 260–268.Google Scholar
7. Johnson, R. E., Quotient rings of rings with zero singular ideals, Pacific J. Math. 11 (1961), 1385–1392.Google Scholar
8. Wu, L. E. T. and Jans, J. P., On quasi-projectives, Illinois J. Math. 11 (1967), 439–448.Google Scholar