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Weakly Projective and Weakly Injective Modules

Published online by Cambridge University Press:  20 November 2018

S. K. Jain ,
S. R. López-Permouth ,
K. Oshiro  and
M. A. Saleh
S. K. Jain
Affiliation:
Ohio University Athens, Ohio 45701 U.S.A.
S. R. López-Permouth
Affiliation:
Ohio University Athens, Ohio 45701 U.S.A.
K. Oshiro
Affiliation:
Yamaguchi University Yoshido, Yamaguchi Japan
M. A. Saleh
Affiliation:
BirZeit University, BirZeit West Bank
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Abstract

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A module M is said to be weakly N-projective if it has a projective cover π: P(M) ↠M and for each homomorphism : P(M) → N there exists an epimorphism σ:P(M) ↠M such that (kerσ) = 0, equivalently there exists a homomorphism :M ↠N such that σ= . A module M is said to be weakly projective if it is weakly N-projective for all finitely generated modules N. Weakly N-injective and weakly injective modules are defined dually. In this paper we study rings over which every weakly injective right R-module is weakly projective. We also study those rings over which every weakly projective right module is weakly injective. Among other results, we show that for a ring R the following conditions are equivalent:

(1) R is a left perfect and every weakly projective right R-module is weakly injective.

(2) R is a direct sum of matrix rings over local QF-rings.

(3) R is a QF-ring such that for any indecomposable projective right module eR and for any right ideal I, soc(eR/eI) = (eR/eJ)n for some positive integer n.

(4) R is right artinian ring and every weakly injective right R-module is weakly projective.

(5) Every weakly projective right R-module is weakly injective and every weakly injective right R-module is weakly projective.

Keywords

16D4016D5016L3016L6016P20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 5 , 01 October 1994 , pp. 971 - 981
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Al-Huzali, A. H., Jain, S. K. and López-Permouth, S. R., Rings whose cyclics have finite Goldie dimension, J. Algebra 153(1992), 37–40.Google Scholar
2. Al-Huzali, A. H., Jain, S. K. and López-Permouth, S. R., On the relative weak injectivity of rings and modules, Non-commutative Ring Theory, Springer-Verlag, New York, Heidelberg 1448, 1990, 93–98.Google Scholar
3. Anderson, F. W. and Fuller, K. R., Rings and categories of modules, Springer-Verlag, New York, Heidelberg, 1974.Google Scholar
4. Clark, John and van Huynh, Dinh, When is a self-injective semiperfect ring Quasi-Frobenius?, preprint.Google Scholar
5. Faith, C., Algebra II, Ring Theory, Springer-Verlag, New York, Heidelberg, 1976.Google Scholar
6. Faith, C., When self-injective rings are Quasi-Frobenius: A report on a problem, Centre de Recerca Matemtica, Institut DàEstudis Catalans, Barcelona, preprint, 1990.Google Scholar
7. Jain, S. K. and López-Permouth, S. R., Rings whose cyclics are essentially embeddable in projectives, J. Algebra 128(1990), 257–269.Google Scholar
8. Jain, S. K., López-Permouth, S. R. and Singh, S., On a class ofQI-rings, Glasgow Math. J. 34(1992), 75–81.Google Scholar
9. Jain, S. K., López-Permouth, S. R. and Saleh, M. A., Weakly projective modules. In: Ring Theory: Proceedings OSU, Denison Conference, World Scientific Press, New Jersey, (1993), 200–208.Google Scholar
10. Kasch, F., Modules and rings, Academic press, London, New York, 1982.Google Scholar
11. López-Permouth, S. R., Rings characterized by their weakly injective modules, Glasgow Math. J. 34(1992), 349–353.Google Scholar
12. Oshiro, K., Lifting modules, extending modules and their applications to QF-rings, Hokkaido Math. J. 13(1984), 310–338.Google Scholar
13. Wu, L. E. T. and Jans, J. P., On quasi projectives, Illinois J. Math. 11(1967), 439–448.Google Scholar