# WEAKLY ⊕-SUPPLEMENTED MODULES AND WEAKLY D2 MODULES

• Hai, Phan The (Department for Management of Science and Technology Development Ton Duc Thang University) ;
• Kosan, Muhammet Tamer (Department of Mathematics Gazi University) ;
• Quynh, Truong Cong (The University of Danang - University of Science and Education)
• 투고 : 2019.04.22
• 심사 : 2020.03.16
• 발행 : 2020.05.31

#### 초록

In this paper, we introduce and study the notions of weakly ⊕-supplemented modules, weakly D2 modules and weakly D2-covers. A right R-module M is called weakly ⊕-supplemented if every non-small submodule of M has a supplement that is not essential in M, and module MR is called weakly D2 if it satisfies the condition: for every s ∈ S and s ≠ 0, if there exists n ∈ ℕ such that sn ≠ 0 and Im(sn) is a direct summand of M, then Ker(sn) is a direct summand of M. The class of weakly ⊕-supplemented-modules and weakly D2 modules contains ⊕-supplemented modules and D2 modules, respectively, and they are equivalent in case M is uniform, and projective, respectively.

#### 참고문헌

1. H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488. https://doi.org/10.2307/1993568
2. V. Camillo, Coherence for polynomial rings, J. Algebra 132 (1990), no. 1, 72-76. https://doi.org/10.1016/0021-8693(90)90252-J
3. S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457-473. https://doi.org/10.2307/1993382
4. R. R. Colby and E. A. Rutter, Jr., Generalizations of QF - 3 algebras, Trans. Amer. Math. Soc. 153 (1971), 371-386. https://doi.org/10.2307/1995563
5. A. Harmanci, D. Keskin, and P. F. Smith, On ${\oplus}$-supplemented modules, Acta Math. Hungar. 83 (1999), no. 1-2, 161-169. https://doi.org/10.1023/A:1006627906283
6. D. V. Huynh, Rings in which no nonzero complement is small, Preprint.
7. D. Keskin, P. F. Smith, and W. Xue, Rings whose modules are ${\oplus}$-supplemented, J. Algebra 218 (1999), no. 2, 470-487. https://doi.org/10.1006/jabr.1998.7830
8. T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, 189, Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-1-4612-0525-8
9. G. Lee, S. T. Rizvi, and C. S. Roman, Dual Rickart modules, Comm. Algebra 39 (2011), no. 11, 4036-4058. https://doi.org/10.1080/00927872.2010.515639
10. W. Li, J. Chen, and F. Kourki, On strongly C2 modules and D2 modules, J. Algebra Appl. 12 (2013), no. 7, 1350029, 14 pp. https://doi.org/10.1142/S0219498813500291
11. S. H. Mohamed and B. J. Muller, Continuous and discrete modules, London Mathematical Society Lecture Note Series, 147, Cambridge University Press, Cambridge, 1990.
12. W. K. Nicholson, Semiregular modules and rings, Canadian J. Math. 28 (1976), no. 5, 1105-1120. https://doi.org/10.4153/CJM-1976-109-2
13. K. Oshiro, Semiperfect modules and quasisemiperfect modules, Osaka Math. J. 20 (1983), no. 2, 337-372. http://projecteuclid.org/euclid.ojm/1200776022
14. G. Puninski and P. Rothmaler, When every finitely generated flat module is projective, J. Algebra 277 (2004), no. 2, 542-558. https://doi.org/10.1016/j.jalgebra.2003.10.027
15. T. C. Quynh, M. T. Kosan, and L. V. Thuyet, On (semi)regular morphisms, Comm. Algebra 41 (2013), no. 8, 2933-2947. https://doi.org/10.1080/00927872.2012.667855
16. T. C. Quynh, M. T. Kosan, and L. V. Thuyet, On Automorphism-Invariant Rings with Chain Conditions, Vietnam J. Math. 48 (2020), no. 1, 23-29. https://doi.org/10.1007/s10013-019-00336-8
17. S. T. Rizvi and C. S. Roman, On direct sums of Baer modules, J. Algebra 321 (2009), no. 2, 682-696. https://doi.org/10.1016/j.jalgebra.2008.10.002
18. S. Singh and A. K. Srivastava, Dual automorphism-invariant modules, J. Algebra 371 (2012), 262-275. https://doi.org/10.1016/j.jalgebra.2012.08.012
19. L. W. Small, Semihereditary rings, Bull. Amer. Math. Soc. 73 (1967), 656-658. https://doi.org/10.1090/S0002-9904-1967-11812-3
20. Y. Utumi, On continuous regular rings, Canad. Math. Bull. 4 (1961), 63-69. https://doi.org/10.4153/CMB-1961-011-6
21. R. B. Warfield, Jr., Decomposability of finitely presented modules, Proc. Amer. Math. Soc. 25 (1970), 167-172. https://doi.org/10.2307/2036549
22. R. Wisbauer, Foundations of Module and Ring Theory, revised and translated from the 1988 German edition, Algebra, Logic and Applications, 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.