jzs-10818

On Cancellation Modules

Adil Kadir Jabbar1 and Dana Jamal Saeed*2

1Department of Mathematics, College of Science, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq

2Directorate of Education of Sulaimani, Iraq

 *Corresponding author’s e-mail: danaj41141@gmail.com

Original: 20 February 2020        Revised: 5 September 2020        Accepted: 26 September 2020        Published online: 20 December 2020  

Doi Link:

Abstract

In this paper, the concept of cancellation modules is introduced, where  is a prime ideal of a commutative ring  with identity, also the effect of localization on cancellation modules is studied and some properties of this type of modules are given, furthermore some conditions are given which make cancellation modules as a cancellation and conversely. Also, the effect of ascending chain conditions and descending chain conditions on the cancellation modules are studied and different characterizations of this type of cancellation modules are given. 

Key Words: Multiplicative system, cancellation ideal, cancellation module, and -cancellation module

References 

[1] M. M. Ali and D. J. Smith. "Pure submodules of Multiplication Modules". Contributions to Algebra and Geometry. Vol. 45, No. 1, pp. 61-74. (2004).

[2] D. D. Anderson and D. F. Anderson. "Some remarks on cancellation ideals". Mathematica Japonica. Vol. 29, No. 6, pp. 879-886. (1984).

[3] S. E. Atani and A. Y. Darani. "Notes on the Primal Submodules". Chiang Mai Journal of Science. Vol. 35, No. 3, pp. 399-410. (2008).

[4] D. M. Burton. "A First Course in Rings and Ideals". Addison- Wesely Publishing Company, Canada, USA. (1970).

[5] R. Gilmer. "Multiplicative Ideal Theory". Marcel Dekker Inc, New York, USA. (1972).

[6] A. K. Jabbar. "On locally Artinian modules". International Journal of Algebra. Vol. 6, No. 27. pp. 1325-1334 (2012).

[7] A. K. Jabbar. "A generalization of prime and weakly prime submodules". Pure Mathematical Sciences. Vol. 2, No. 1, pp. 1-11. (2013).

[8] C. U. Jensen. "Arithmetical Rings". Acta Mathematica Academiae Scientiarum Hungaricae. Vol. 17, No. 1-2, pp. 115-123. (1966).

[9] D. M. Larsen and P. J. McCarthy. "Multiplicative Theory of Ideals". Academic Press, New York and London. (1971).

[10] A. S. Mijbass. "On Cancellation Module". M.Sc. Thesis, University of Baghdad. (1992).

[11] A. G. Naoum and A. S. Mijbass. "Weak Cancellation Modules". Kyungpook Math. J. Vol. 37, pp. 73-82. (1997).

[12] B. N.  Shihab. "On Restricted Cancellation Module". M.Sc. Thesis, University of Tikrit. (2000).

[13] R. Wisbauer. "Foundations of Module and Ring Theory". Gordon and Breach Science Publishers, Germany, (1991).

View Full Article