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On a class of QI-rings

Published online by Cambridge University Press:  18 May 2009

S. K. Jain ,
S. R. López-Permouth  and
Surjeet Singh
S. K. Jain
Affiliation:
Department of Mathematics, Ohio University, Athens, OH 45701
S. R. López-Permouth
Affiliation:
Department of Mathematics, Ohio University, Athens, OH 45701
Surjeet Singh
Affiliation:
Department of Mathematics, Kuwait University, Kuwait
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The concept of weak relative-injectivity of modules was introduced originally in [10], where it is shown that a semiperfect ring R is such that every cyclic right module is embeddable essentially in a projective right R-module if and only if R is right artinian and every indecomposable projective right R-module is uniform and weakly R-injective. We show that in the above characterization the requirement that indecomposable projective right R-modules be uniform is superfluous (Theorem 1.11). In this paper we further the study of weak relative-injectivity by considering the class of rings for which every right module is weakly injective relative to every finitely generated right module. We refer to such rings as right weakly-semisimple rings. The class of right weakly-semisimple rings includes properly all semisimple rings and is a subclass of the class of right QI-rings. A ring R is said to be a right QI-ring if every quasi-injective right R-module is injective. QI-rings have been studied in [2], [3], [4], [6], [7], [8], [11], among others.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 1 , January 1992 , pp. 75 - 81
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

REFERENCES

1.Beachy, J. and Blair, W. D., Rings whose faithful left ideals are cofaithful, Pacific J. Math. 58 (1975), 1–13.CrossRefGoogle Scholar
2.Boyle, A. K., Ql-rings, Hereditary, Trans. Amer. Math. Soc. 192 (1974), 115–120.Google Scholar
3.Boyle, A. K. and Goodearl, K. R., Rings over which certain modules are injective, Pacific J. Math. 58 (1975), 43–53.CrossRefGoogle Scholar
4.Boyle, A. K., Injectives containing no proper quasi-injective submodules, Comm. Algebra 4 (1976), 775–785.CrossRefGoogle Scholar
5.Eisenbud, D. and Robson, J. C., Modules over Dedekind prime rings, J. Algebra 16 (1970), 67–85.CrossRefGoogle Scholar
6.Faith, C., When are proper cyclics injective? Pacific J. Math. 45 (1973), 97–111.CrossRefGoogle Scholar
7.Faith, C., On hereditary rings and Boyle's conjecture, Arch. Math. (Basel) 27 (1976), 113–119.CrossRefGoogle Scholar
8.Golan, J. S. and Papp, Z., Cocritically nice rings and Boyle's conjecture, Comm. Algebra 8 (1980), 1775–1798.CrossRefGoogle Scholar
9.Gordon, R. and Robson, J. C., Krull dimension, Mem. Amer. Math. Soc. 133 (1970).Google Scholar
10.Jain, S. K. and López-Permouth, S. R., Rings whose cyclics are essentially embeddable in projective modules, J. Algebra 128 (1990), 257–269.CrossRefGoogle Scholar
11.Kosler, K., On hereditary rings and noetherian V-rings, Pacific J. Math. 103 (1982), 467–473.CrossRefGoogle Scholar
12.Michler, G. O. and Villamayor, O. E., On rings whose simple modules are injective, J. Algebra 25 (1973), 185–201.CrossRefGoogle Scholar
13.Mohamed, S. and Miiller, B. J., Continuous and discrete modules, London Math. Soc. Lecture Note Series 147 (Cambridge University Press, 1990).CrossRefGoogle Scholar