Issues‎ > ‎Vol 20n3-4‎ > ‎


On Primarily Hollow Module

 Adil Kadir Jabbar1 & Payman Mahmood Hamaali2

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


An R- module  M is called primarily hollow module if every primary submodule of  is a small submodule in. This definition extends several notions in the literature of hollow module. Some certain equivalent conditions to primarily hollow modules. Also, we verified that a primarily hollow module which contains a maximal submodule is a multiplication module and every multiplication primarily hollow module is indecomposable.

Key Words: Primary submodule, primarily hollow module, multiplication module(submodule)



[1] El-Bast, Z. A. and Smith, P. P., "Multiplication modules". Communications in Algebra, Vol. 16, No. 4, pp. 755-779. (1988).

[2] Fleury, P., "Hollow modules and local endomorphism rings". Pacific Journal of Mathematics, Vol. 53, No. 2, pp. 379-385. (1974).

[3] Hamaali, P. and Al-Hashimi. B., "Hollow Modules and Semihollow Modules", Journal of Zankoi Sulaimani, Vol. 19. (2017).

[4] Nikseresht, A. and Sharif, H., "Fully Primary Modules and some Variations". Journal of Algebra and Related Topics, pp. 1-17. (2013).

[5] EL-Atrash, M. S.  and Ashour, A. E., "On Primary Compactly Packed Bezout Modules, Department of Mathematics, College of Science, Islamic University of Gaza, Gaza Palestine, pp. 27-28. (2002).

[6] Atani, S. E., Çallıalp, F., and Tekir, Ü. "A short note on the primary submodules of multiplication modules". International Journal of Algebra, Vol. 1, No. 8, pp. 381-384. (2007).

[7] Marcelo, A., Marcelo, F. and Rodríiguez, C., "Some results on prime and primary submodules". Proyecciones (Antofagasta), Vol. 22, No.3, pp. 201-208. (2003).

[8] Bataineh, M., and Kuhail, S. "Generalizations of primary ideals and submodules". International Journal of Contemporary Mathematical Sciences, Vol. 6, No. 17-20, pp. 811-824. (2011).

[9] Rangaswamy, K. M. "Modules with finite spanning dimension". Canad. Math. Bull, Vol. 20, No. 2, pp. 255-26. (1977).  

[10] Tutuncu, D. K., "On Coclosed Submodules", Indian J. Pure appl. Math, Vol. 36, No. 3, pp. 135-144. (2005).

[11] Bland, P. E. "Rings and their modules". Walter de Gruyter. (2011). 

[12] Ware, R., "Endomorphism rings of projective modules". Transactions of the American Mathematical Society, Vol. 155, No. 1, pp. 233-256. (1971).

[13] Azizi, A., and Jayaram, C., "Some applications of the product of submodules in multiplication modules". Iranian Journal of Science and Technology (Sciences), Vol. 35, No. 4, pp. 273-277. (2011).

[14] Ebrahimpour, M., On generalisations of almost prime and weakly prime ideals. Bulletin of the Iranian Mathematical Society, Vol. 40, No. 2, pp. 531-540. (2014).

[15] Ebrahimpour, M., and Nekooei, R., "On Generalization of prime submodules". Bulletin of the Iranian Mathematical Society, Vol. 39, No. 5, pp. 919-939. (2013).

[16] Varadarajan, K., "Dual Goldie dimension", Comm. Algebra, Vol. 7, No. 6, pp. 565–610. (1979).

[17] Reiter, E., "A dual to the Goldie ascending chain condition on direct sums of submodules", Bull. Calcutta Math. Soc., Vol. 73, pp. 55–63. (1981).

[18] Grezeszcuk, P. and Puczylowski, E., "On Goldie and dual Goldie dimension", Journal of Pure and Applied Algebra, Vol. 31, pp. 47–55. (1984).

[19] Zhao, R., "Uniform and couniform dimension of generalized inverse polynomial modules". Bulletin of the Korean Mathematical Society, Vol. 49, No. 5, pp. 1067-1079. (2012).