Advances in Ring Theory pp 221-238 | Cite as

# Weak Relative Injective *M*-Subgenerated Modules

Conference paper

## Abstract

We study weak relative injective and relative tight modules in the category *σ*[*M*], where *М is* a right R-module. Many of the known results in the category of right R-modules are extended to *σ*[*М*]without assuming either *М* is projective or finitely generated. Conditions are given for a *A*-tight module to be weakly *A*-injective in *σ*[*M*]*.* Modules for which every submodule is weakly injective (tight) in *σ*[*М*] are characterized. Modules *М* for which every module in *σ*[*М*] is weakly injective and for which weakly injective modules are closed under direct sums are studied.

## Keywords

Injective Module Cyclic Module Indecomposable Module Essential Extension Injective Hull
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## References

- 1.A.H. Al-Huzali, S.K. Jain and S.R. López-Permouth,
*Rings whose cyclics have finite Goldie dimension*, J. Algebra**153**(1992), 37–40.MathSciNetMATHCrossRefGoogle Scholar - 2.A.H. Al-Huzali, S.K. Jain and S.R. López-Permouth,
*On the weak relative injectivity of rings and modules*, Lecture notes in Math.**1448**(1990), 93–98.CrossRefGoogle Scholar - 3.G.M.Brodskii, M.Saleh, Le Van Thuyet and R.Wisbauer,
*On weak injectivity of direct sum of modules*,to be published.Google Scholar - 4.V.P. Camillo,
*Modules whose quotients have finite Goldie dimension*, Pacific J. Math.,**69**(1977), 337–338.MathSciNetMATHCrossRefGoogle Scholar - 5.N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer,
*“Extending modules”*, Pitman Research Notes in Mathematics series, Longman, Harlow, 1994.Google Scholar - 6.S.K. Jain and S.R. López-Permouth,
*A survey on theory of weakly injective modules*, Computational Algebra, Lecture notes in Pure and applied Math.**151**, Dekker, New York (1994), 205–232Google Scholar - 7.S.K. Jain and S.R. López-Permouth,
*Weakly injective modules over hereditary noetherian prime rings*, J. Australian Math. Soc., (series A),**58**(1995), 287–297.MATHCrossRefGoogle Scholar - 8.S.K. Jain, S.R. López-Permouth and S. Singh,
*On a class of QI rings*, Glasgow Math. J.**34**(1992), 75–81.MathSciNetMATHCrossRefGoogle Scholar - 9.S.R. López-Permouth,
*Rings characterized by their weakly injective modules*, Glasgow Math. J.**34**(1992), 349–353.MathSciNetMATHCrossRefGoogle Scholar - 10.S.R. López-Permouth, S.Tariq Rizvi and M.F. Yousif,
*Some characterisations of semiprme Goldie rings*, Glasgow Math. J.**35**(1993), 357–365.MathSciNetMATHCrossRefGoogle Scholar - 11.S.R. López-Permouth, and S.Tariq Rizvi,
*On certain classes of QI-rings*, Methods in Module Theory (Colorado Springs C.O 1991), Lecture notes in pure and applied Math.**140**, Dekker, New York, 1993, 227–235.Google Scholar - 12.S.M. Mohamed and B.J. Müller,
*“Continuous and discrete modules”*,London Math. Soc. Lecture Notes Series**147**, Cambridge, 1990.CrossRefGoogle Scholar - 13.N. Vanaja,
*All finitely generated M -subgenerated modules are extending*, Comm. Algebra,**24**(1996), 543–578.MathSciNetMATHCrossRefGoogle Scholar - 14.R. Wisbauer,
*“Foundations of module and ring theory”*, Gordon and Breach, Reading, 1991.Google Scholar - 15.Y. Zhou,
*Weak injectivity and module classes*, to be published.Google Scholar - 16.Y. Zhou,
*Notes on weakly-semisimple rings*, Bull. Austr. Math. Soc.,**53**(1996), 517–525.CrossRefGoogle Scholar

## Copyright information

© Springer Science+Business Media New York 1997