Some Properties of Faithful Locally Multiplication
Modules
Adil Kadir Jabbar1*
, Payman Mahmood Hamaali2 & Adham Idris Abid3
Abstract
1,2 Department of Mathematics, College of Science, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq
3Department of Mathematics, College of Science Education, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq
*Corresponding author Email: adil.jabbar@univsul.edu.iq
Abstract
An 𝑅 −module 𝑀 is called a multiplication module if each submodule of 𝑀 is a product
of an ideal of 𝑅 by the module and it is called a faithful module if 𝑎𝑛𝑛(𝑀) = 0. 𝑀 is
called a locally multiplication module if 𝑀𝑃 is a multiplication module for each prime
ideal 𝑃 of 𝑅 (due to the local ring 𝑅𝑃 on which the module 𝑀𝑃 is defined). In this paper
we explore some new properties of faithful locally multiplication modules.
Key Words:
multiplication
module, faithful
module, locally
multiplication
module, prime
submodule.
References
References
[1] Atani, S. E. and Darani, A. Y.: Notes on the Primal Submodules, Chiang Mai J. Sci. 2008; 35(3), 399-410.
[2] Atani, S. E. and Farzalipour , F. : On Weakly Prime Submodules, Tamkang Journal of Mathematics, Vol.
38, No. 3, 2007, 247-252.
[3] Atiyah, M. F. and Macdonald, I. G.: Introduction to commutative algebra, Addison-wesley publishing
company, 1969.
[4] Burton , D. M.: A First Course in Rings and Ideals, Addison-Wesely Publishing Company, 1970.
[5] Darani, A. Y. : Almost Primal Ideals in Commutative Rings, Chiang Mai J. Sci. 2011; 38(2) : 161-165.
[6] Jabbar, A. K. and Hasan, N. H.: On Locally Multiplication Modules, International mathematical Forum,
Vol. 11, 2016, no. 5, 213-226.
[7] Jabbar, A. K. and Hasan, N. H.: Some Results Concerning Localization of Commutative Rings and
Modules, International Journal of Algebra, Vol. 9, 2015, no. 8, 403-412.
[8] Jabbar, A. K. and Hussein, P. K.: On Weakly Pure Submodules of Locally Multiplication Modules, JZS-A,
Volume 22, Issue 2, 2020, 353-360.
[9] Jabbar, A. K.: A generalization of prime and weakly prime submodules, Pure Mathematical Sciences, Vol.
2, 2013, no. 1, 1-11.
[10] Jabbar, A. K.: On Locally Artinian Modules, International Journal of Algebra, Vol. 6, 2012 No. 27, 1325-
1334.
[11] Keskin, D. , Ozcan, A. C. and Tiras, Y. : On Prime submodules, Far East J. Math. Sci. 4(2) (1996), 163-
168.
[12] Larsen, M. D. and McCarthy, P. J.: Multiplicative theory of ideals, Academic Press, New York and
London, 1971.
[13] Lee, D. D. and Lee, H. B.: Some Remarks on Faithful Multiplication Modules, Journal of the
Chungcheong Mathematical Society, Vol. 6, 1993, 131-137.
[14] Smith, P. F.: Some remarks on multiplication modules, Arch. Math. 50 (1988), 223-235.
[15] Wisbauer, R. : Foundations of Module and Ring Theory, Gordon and Breach Science Publishers, 1991
