Document Type : Original Manuscript


Department of Mathematics, Faculty of Sciences, University of Mohammed First, Oujda, Morocco


In this paper, we introduce and study the concept of Jacobson monoform modules which
is a proper generalization of that of monoform modules. We present a characterization of semisimple
rings in terms of Jacobson monoform modules by proving that a ring $R$
is semisimple if and only if every $R$-module is Jacobson monoform. Moreover, we demonstrate that over a ring $R$, the properties monoform, Jacobson monoform, compressible, uniform and weakly co-Hopfian are all equivalent.


1. M. Barry and P. C. Diop, Some properties related to commutative weakly FGIrings, JP Journal of algebra, number theory and application, 19(2) (2010), 141– 153.
2. P. C. Diop and A. D. Diallo, Modules whose partial endomorphisms have a δ-small kernels, Proyecciones (Antofagasta), 39(4) (2020), 945–962.
3. A. Haghany and M. R. Vedadi, Modules whose injective endomorphisms are essential, J. Algebra, 243(2) (2001), 765–779.
4. M. A. Inaam Hadi and K. H. Marhoon, Small monoform modules, Ibn ALHaitham Journal For Pure and Applied Sciences, 27(2) (2014), 229–240.
5. A. Kabban and K. Wasan, On Jacobson-small submodules, Iraqi Journal of Science, 60(7) (2019), 1584–1591.
6. T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag New York, 1999.
7. A. El Moussaouy, M. Khoramdel, A. R. Moniri Hamzekolaee and M. Ziane, Weak Hopfcity and singular modules, Annali dell’Universita di Ferrara, 68 (2022), 69– 78.
8. A. El Moussaouy and M. Ziane, Modules in which every surjective endomorphism has a µ-small kernel, Annali dell’Universita di Ferrara, 66 (2020), 325–337.
9. A. El Moussaouy and M. Ziane, Notes on generalizations of Hopfian and coHopfian modules, Jordan J. Math. Stat., 15(1) (2022), 43–54.
10. A. El Moussaouy, M. Ziane and A. R. Moniri Hamzekolaee, Jacobson Hopfian modules, Algebra Discrete Math., 33(1) (2022), 116–127.
11. V. S. Rodrigues and A. A. Santana, A note on a problem due to Zelmanowitz, Algebra Discrete Math., 3 (2009), 85–93.
12. P. F. Smith, Compressible and related modules, In: Pat Goeters and Overtoun M. G. Jenda, (Eds), Abelian groups, rings, modules and homological algebra, Lect. Notes Pure Appl. Math., 249 (2006), 295–313.
13. Y. Talebi, A. R. Moniri Hamzekolaee, A. Harmanci and B. Ungor, Rings for which every cosingular module is discrete, Hacet. J. Math. Stat, 49(5) (2020), 1635–1648.
14. Y. Talebi, A. R. Moniri Hamzekolaee, M. Hosseinpour, A. Harmanci and B. Ungor, Rings for which every cosingular module is projective, Hacet. J. Math. Stat, 48(4) (2019), 973–984.
15. Y. Talebi and N. Vanaja, The torsion theory cogenerated by M-small modules, Comm. Algebra, 30(3) (2002), 1449–1460.
16. J. M. Zelmanowitz, Representation of rings with faithful polyform modules, Comm. Algebra, 14(6) (1986), 1141–1169.
17. Y. Zhou, Generalizations of perfect, semiperfect, and semiregular rings, Algebra Colloq., 7(3) (2000), 305–318.