Issues‎ > ‎Vol19n2‎ > ‎

Hollow and Semihollow Modules


Payman Mahmood Hama Ali & Basil A. Al-Hashimi

University of Sulaimani, College of Science, Department of Mathematics.
University of Baghdad, College of Science, Department of Mathematics.


DOI: https://doi.org/10.17656/jzs.10624

Abstract

Let be an associative ring with identity and be a non-zero unitary left module over . is called a hollow (semihollow) module if every proper (finitely generated proper) submodule of is a small submodule of . The purpose of this work is, to give a comprehensive study of hollow modules and semihollow modules. Moreover, we study the class of modules with finite spanning dimension. We supply the details of the proofs for almost all the results and we illustrate the concepts by examples. Also, we add some results that seem to be new to the best of our knowledge.


Key Words: Hollow module, Semihollow Module, Small Submodules.


References:

[1] F. W. Anderson and K. R. Fuller, "Rings and Categories of Modules", Springer-Verlag, New York, (1992).

[2] N. V. Dung, D. V. Huynh, P. F. Smith and R wisbauer, "Extending Modules", Number 313 in Putman

Research Notes in Mathematics series, Longman Harlow (1994).

[3] P. Fleury, "Hollow Modules and Local Endomorphism Rings", Pac. J. Math, Vol. 53, pp. 379-385, (1974).

[4] P. Fleury, "A Note on Dualizing Goldie Dimension", Canada.Math. Bull. Vol. 17, No. 4, pp. 511-517, (1974).

[5] J. S. Golan "Quasi-Semiperfect Modules Quart", J. Math. Oxford , Vol. 22, No. 2, pp. 173-182, (1971).

[6] N. Hamada and B. Al-Hashimi "Some Results on the Jacobson Radical and the M- Radicals", Abhath Al-

Yarmouk, Vol. 11, No. 2A, pp. 573-579, (2002).

[7] A.L Hamdouni, "On Lifting Modules", Thesis College of science, University of Baghdad, (2001).

[8] D.V. Huynh, "A Note on Quasi-Frobenius Rings", American Math. Society, Vol. 124, No. 2, pp. 371-375, (1996).

[9] T. Inoue, "Sum of Hollow Modules", Osaka J. Math. Vol. 20, pp. 331-336, (1983).

[10] F. Kasch, "Modules and Rings", Academic Press Inc. London, (1982).

[11] P. Hama Ali, "Hollow Modules and Semihollow Modules", Thesis College of science, University of

Baghdad, (2006).

[12] C. Lomp and A. J. Pena, "A Note on Prime Modules", Divulgaciones Matematices, Vol. 8, No. 1, pp. 31-42, (2000).

[13] S. H. Mohamed and B. J. Muller, "Continuous and Discrete Modules", London math. Soc. LNS 147

Cambridge Univ. press, Cambridge, (1990).

[14] K. M. Rangaswamy, "Modules with Finite spanning Dimension", Canada Math. Bull. Vol. 20, No. 2, pp. 255- 262, (1977).

[15] R. Y. Sharp, "Steps in Commutative Algebra", Cambridge University Press, (1990).

[16] R. Wisbauer, "Foundations of Module and Ring Theory", Gordon and Breach Reading (1991).

[17] S. M. Yasean, "Coquasi-Dedekind Modules", PH. D. Thesis, College of science University of Baghdad,

(2003).

[18] H. Zoschinger, "Minimax-Modules", J. Algebra, Vol. 102, pp. 1-32, (1986).