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On a class of QI-rings

  • S. K. Jain (a1), S. R. López-Permouth (a1) and Surjeet Singh (a2)

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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.

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References

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1.Beachy, J. and Blair, W. D., Rings whose faithful left ideals are cofaithful, Pacific J. Math. 58 (1975), 113.
2.Boyle, A. K., Ql-rings, Hereditary, Trans. Amer. Math. Soc. 192 (1974), 115120.
3.Boyle, A. K. and Goodearl, K. R., Rings over which certain modules are injective, Pacific J. Math. 58 (1975), 4353.
4.Boyle, A. K., Injectives containing no proper quasi-injective submodules, Comm. Algebra 4 (1976), 775785.
5.Eisenbud, D. and Robson, J. C., Modules over Dedekind prime rings, J. Algebra 16 (1970), 6785.
6.Faith, C., When are proper cyclics injective? Pacific J. Math. 45 (1973), 97111.
7.Faith, C., On hereditary rings and Boyle's conjecture, Arch. Math. (Basel) 27 (1976), 113119.
8.Golan, J. S. and Papp, Z., Cocritically nice rings and Boyle's conjecture, Comm. Algebra 8 (1980), 17751798.
9.Gordon, R. and Robson, J. C., Krull dimension, Mem. Amer. Math. Soc. 133 (1970).
10.Jain, S. K. and López-Permouth, S. R., Rings whose cyclics are essentially embeddable in projective modules, J. Algebra 128 (1990), 257269.
11.Kosler, K., On hereditary rings and noetherian V-rings, Pacific J. Math. 103 (1982), 467473.
12.Michler, G. O. and Villamayor, O. E., On rings whose simple modules are injective, J. Algebra 25 (1973), 185201.
13.Mohamed, S. and Miiller, B. J., Continuous and discrete modules, London Math. Soc. Lecture Note Series 147 (Cambridge University Press, 1990).

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On a class of QI-rings

  • S. K. Jain (a1), S. R. López-Permouth (a1) and Surjeet Singh (a2)

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