ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 15 (2020), 281 -- 293
This work is licensed under a Creative Commons Attribution 4.0 International License.
SYMMETRY OF EXTENDING PROPERTIES IN NONSINGULAR UTUMI RINGS
Thuat Do, Hai Dinh Hoang, Truong Dinh Tu
Abstract. This paper presents the right-left symmetry of the CS and max-min CS conditions on nonsingular rings, and generalization to nonsingular modules. We prove that a ring is right nonsingular right CS and left Utumi if and only if it is left nonsingular left CS and right Utumi. A nonsingular Utumi ring is right max (resp. right min, right max-min) CS if and only if it is left min (resp. left max, left max-min) CS. In addition, a semiprime nonsingular ring is right max-min CS with finite right uniform dimension if and only if it is left max-min CS with finite left uniform dimension.
2020 Mathematics Subject Classification: 16D70, 16S50
Keywords: CS modules, max-min CS rings, nonsingular rings, semiprime rings, Utumi rings
References
K. R. Goodearl. Ring Theory: Nonsingular Rings and Modules. Pure and Applied Mathematics, No. 33. Marcel Dekker, Inc., New York-Basel, 1976. MR0429962.
D. V. Huynh, S. K. Jain, and S. R. Lopez-Permouth. On the symmetry of the Goldie and CS conditions for prime rings. Proc. Amer. Math. Soc. 128:11(2000), 3153-3157. MR1670375. Zbl 112616009.
D. V. Huynh. The symmetry of the CS condition on one-sided ideals in a prime ring. J. Pure and Applied Algebra, 212 (2008), 9-13. MR2355030. Zbl 112616009.
S. K. Jain, Husain S. Al-Hazmi, and Adel N. Alahmadi. Right-Left Symmetry of Right Nonsingular Right Max-Min CS Prime Rings. Communications in Algebra, 34 (2006), 3883-3889. MR2267561. Zbl 112616010.
R. E. Johnson. Quotient rings with zero singular ideal. Pacific J. Math. 11 (1961), 1358-1392. MR143779. Zbl 020404504.
S. M. Khuri. Endomorphism rings of nonsingular modules. Ann. Sci. Math. Quebec, 4 (1980), 145-152. MR599052. Zbl 045116021.
S. M. Khuri. Modules whose endomorphism rings have isomorphic maximal left and right quotient rings. Proc. Amer. Math. Soc. 85:2 (1982), 161-164. MR652433. Zbl 050116003.
S. M. Khuri. Correspondence theorems for modules and their endomorphism rings. J. Algebra, 122(1989), 380-396. MR999081. Zbl 067916023.
S. M. Khuri. Nonsingular retractable modules and their endomorphism rings. Bull. Austral. Math. Soc. 43(1991), 63-71. MR1086718. Zbl 071916004.
D. V. Thuat, H. D. Hai and N. V. Sanh. On Goldie prime CS-modules. East-West J. Math. 16:2 (2014), 131-140. MR3410764. Zbl 133416020.
Y Utumi. On prime J-rings with uniform one-sided ideals. Amer. J. Math. 85:4 (1963), 583-596. MR174591. Zbl 014728603.
Truong Dinh Tu (corresponding author)
NLP-KD Lab,
Faculty of Information Technology,
Ton Duc Thang University,
Ho Chi Minh City, Vietnam.
e-mail: truongdinhtu@tdtu.edu.vn
Hai Dinh Hoang
International Cooperation Office,
Hong Duc University,
565 Quang Trung St, Dong Ve ward, Thanh Hoa city, Vietnam.
e-mail: hoangdinhhai@hdu.edu.vn
Thuat Do
Institute of Research and Development,
Duy Tan University,
Da Nang 550000, Vietnam.
and
Department of Science and Technology,
Nguyen Tat Thanh University,
Ho Chi Minh City, Vietnam.
e-mail: thuat86@gmail.com