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Issue: Special Issue (10.2026)
Section: Các bài báo
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Baupradist , S., Sangchan , P. ., & Hoang, D. H. (2026). ON RIGHT AND LEFT QUASI-MONO-PRINCIPALLY INJECTIVE MODULES AND THEIR ENDOMORPHISM RINGS. Hong Duc University Journal of Science, 10. https://hdujs.edu.vn/index.php/js/article/view/1589

ON RIGHT AND LEFT QUASI-MONO-PRINCIPALLY INJECTIVE
MODULES AND THEIR ENDOMORPHISM RINGS 

Samruam  Baupradist 1, Poramate Sangchan 1, Dinh Hai Hoang2,
1 Chulalongkorn University
2 Hong Duc University

Main Article Content

Abstract

Let M be a right R-module and let S = EndR(M). We study two classes of modules arising from a modification of relative principal injectivity, namely right and left M-mono-principally injective modules. A right R-module N is called right M-mono-principally injective if, for every 0 ≠ s ∈ S, there exists a monomorphism t in S such that every R-homomorphism from ts(M) to N extends to an R-homomorphism from M to N. The left-handed version is defined analogously by considering st(M). We establish several properties and characterizations of these modules and investigate their connections with the endomorphism ring of M. In particular, we obtain consequences concerning local endomorphism rings, uniform modules, directly finite and co-Hopfian modules, and the GC2 condition.
Corresponding results for rings are obtained by taking M = RR.

Keywords

M-mono-principally injective module, quasi-mono-principally injective module, endomorphism ring, uniform module, GC₂ condition

Article Details

Author Biographies

Assoc. Prof Samruam  Baupradist , Chulalongkorn University

Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

Poramate Sangchan, Chulalongkorn University

Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

Dr. Dinh Hai Hoang, Hong Duc University

Faculty of Natural Sciences, Hong Duc University, 565 Quang Trung Str., Hac Thanh Ward, Thanh Hoa Province, Viet Nam

References

[1] V. P. Camillo, Modules whose quotients have finite Goldie dimension, Pacific J.
Math. 69(2) (1977), 337–338. [2] F. Kasch, Moduln und Ringe, Mathematische Leitfa¨den, Vieweg, Wiesbaden, 1977. [3] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd ed., Graduate Texts in Mathematics 13,Springer-Verlag, New York, 1992. [4] H. Q. Dinh and D. V. Huynh, Some results on self-injective rings and Sigma-CS rings, Comm. Algebra 31(12) (2003), 6063–6077. [5] S. H. Mohamed and B. J. Mu¨ller, Continuous and Discrete Modules, London Mathematical Society Lecture Note Series 147, Cambridge University Press, Cambridge, 1990. [6] R. Wisbauer, Foundations of Module and Ring Theory, Algebra, Logic and Applications 3,Gordon and Breach Science Publishers, 1991. [7] 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. [8] J. Clark and P. F. Smith, On semi-artinian modules and injectivity conditions, Proc.
Edinburgh Math. Soc. 39(2) (1996), 263–270. [9] N. V. Sanh, K. P. Shum, S. Dhompongsa and S. Wongwai, On quasi-principally injective modules, Algebra Colloq. 6(3) (1999), 269–276. [10] H. D. Hai, N. V. Sanh and A. Sudprasert, A weaker form of p-injectivity, Southeast Asian Bull. Math. 33 (2009), 1063–1069. [11] A. Sudprasert, H. D. Hai, S. Sanpinij and N. V. Sanh, Endomorphism Ring of Quasi-rp-injective and Quasi-lp-injective Modules, East-West J. Math. 11(2) (2009), 231–240.
[12] Z. Zhu, Some results on GQP-injective modules, Int. Electron. J. Algebra 5 (2009), 1–6.
[13] Y. Zhou, Rings in which certain right ideals are direct summands of annihilators, J. Aust. Math. Soc. 73(3) (2002), 335–346.