On a generalization of $C_2$-modules

Year 2022, Volume: 51 Issue: 2, 430 - 442, 01.04.2022

Abdoul Djibril Diallo Papa Cheikhou Dıop , Rachid Tribak

https://doi.org/10.15672/hujms.792212

Cited By: 1

Abstract

A module $M$ is called a $C_{21}$-module if, whenever $A$ and $B$ are submodules of $M$ with $A \cong B$, $A$ is nonsingular and $B$ is a direct summand of $M$, then $A$ is a direct summand of $M$. Various examples of $C_{21}$-modules are presented. Some basic properties of these modules are investigated. It is shown that the class of rings $R$ over which every $C_{21}$-module is a $C_2$-module is exactly that of right SI-rings. Also, we prove that for a ring $R$, every $R$-module has $(C_{21})$ if and only if $R$ is a right t-semisimple ring.

Keywords

$C_{21}$-module , $C_2$-module , nonsingular module , singular module , SI-ring , t-semisimple ring

References

• [1] I. Amin, Y. Ibrahim and M. Yousif, C3-modules, Algebra Colloq. 22 (4), 655-670, 2015.
• [2] Sh. Asgari, T-continuous modules, Comm. Algebra, 45 (5), 1941-1952, 2017.
• [3] Sh. Asgari, T-quasi-continuous modules, Comm. Algebra, 47 (5), 1939-1953, 2019.
• [4] Sh. Asgari and A. Haghany, t-Extending modules and t-Baer modules, Comm. Alge- bra, 39 (5), 1605-1623, 2011.
• [5] Sh. Asgari, A. Haghany and Y. Tolooei, T-semisimple modules and T-semisimple rings, Comm. Algebra, 41 (5), 1882-1902, 2013.
• [6] V. Camillo, Y. Ibrahim, M. Yousif and Y. Zhou, Simple-direct-injective modules, J. Algebra, 420, 39-53, 2014.
• [7] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules. Supplements and Projectivity in Module Theory, Frontiers in Mathematics, Birkhäuser, Basel, 2006.
• [8] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending Modules, Pitman Research Notes in Mathematics series 313, Longman Scientific & Technical, Harlow, 1994.
• [9] L. Fuchs, Infinite Abelian Groups, Vol. I, Academic Press, New York, 1970.
• [10] L. Fuchs, Infinite Abelian Groups, Vol. II, Academic Press, New York, 1973.
• [11] K.R. Goodearl, Ring Theory. Nonsigular Rings and Modules, Marcel Dekker, New York, 1976.
• [12] D.K. Harrison, Infinite Abelian groups and homological methods, Ann. of Math. (2) 69 (2), 366-391, 1959.
• [13] L. Jeremy, Sur les modules et anneaux quasi-continus, C. R. Acad. Sci. Paris (Série A) 273, 80-83, 1971.
• [14] I. Kaplansky, Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72, 327-340, 1952.
• [15] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.
• [16] F. Kourki, When maximal linearly independant subsets of a free module have the same cardinality?, in: Modules and Comodules, Trends in Mathematics, 281-293, Birkhäuser, Verlag, Basel, Switzerland, 2008.
• [17] T.Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999.
• [18] G. Lee, S.T. Rizvi and C.S. Roman, Dual Rickart modules, Comm. Algebra, 39 (11), 4036-4058, 2011.
• [19] G. Lee, C.S. Roman and X. Zhang, Modules whose endomorphism rings are division rings, Comm. Algebra, 42 (12), 5205-5223, 2014.
• [20] S.H. Mohamed and T. Bouhy, Continuous modules, Arabian J. Sci. Eng. 2, 107-122, 1977.
• [21] S.H. Mohamed and B.J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Note Series 147, Cambridge University Press, Cambridge, 1990.
• [22] W.K. Nicholson and M.F. Yousif, Quasi-Frobenius Rings, Cambridge University Press, Cambridge, 2003.
• [23] F.L. Sandomierski, Semisimple maximal quotient rings, Trans. Amer. Math. Soc. 128, 112-120, 1967.
• [24] P.F. Smith and A. Tercan, Generalizations of CS-modules, Comm. Algebra 21 (6), 1809-1847, 1993.
• [25] T. Takeuchi, On direct modules, Hokkaido Math. J. 1 (2), 168-177, 1972.
• [26] A. Tuganbaev, Rings Close to Regular, Mathematics and Its Applications, vol. 545, Kluwer Academic Publishers, Dordrecht, 2002.
• [27] Y. Utumi, On continuous regular rings, Canad. Math. Bull. 4 (1), 63-69, 1961.
• [28] R. Ware, Endomorphism rings of projective modules, Trans. Amer. Math. Soc. 155 (1), 233-256, 1971.
• [29] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publishers, Philadelphia, 1991.