Rings such that, for each unit u, u − u n belongs to the Jacobson radical

Year 2020, Volume: 49 Issue: 4, 1397 - 1404, 06.08.2020

M. Tamer Koşan , Truong Cong Quynh , Tülay Yıldırım , Jan žemlička

https://doi.org/10.15672/hujms.542574

Cited By: 3

Abstract

A ring R is said to be n-UJ if u − u n ∈ J(R) for each unit u of R, where n > 1 is a fixed integer. In this paper, the structure of n-UJ rings is studied under various conditions. Moreover, the n-UJ property is studied under some algebraic constructions. Mathematics Subject Classification (2010). 16N20, 16D60, 16U60, 16W10

Keywords

UU-ring, UJ-ring, unit, Jacobson radical, clean ring, (semi)regular ring, reduced ring

References

• [1] D.D. Anderson, D. Bennis, B. Fahid and A. Shaiea, On n-trivial extensions of rings, Rocky Mountain. J. Math. 47, 2439–2511, 2017.
• [2] A. Badawi, On abelian π-regular rings, Comm. Algebra, 25 (4), 1009–1021, 1997.
• [3] G.F. Birkenmeier, H.E. Heatherly, and E.K. Lee, Completely prime ideals and associated radicals, Proc. Biennial Ohio State-Denison Conference 1992, edited by S. K. Jain and S. T. Rizvi, World Scientific, Singapore-New Jersey-London-Hong Kong, 102–129, 1993.
• [4] G. Calugareanu, UU rings, Carpathian J. Math. 31 (2), 157–163, 2015.
• [5] J. Cui and X. Yin, Rings with 2-UJ property, Comm. Algebra, 48 (4), 1382–1391, 2020.
• [6] P. Danchev, Rings with Jacobson units, Toyama Math. J. 38, 61–74, 2016.
• [7] P. Danchev and T.Y. Lam, Rings with unipotent units, Publ. Math. Debrecen, 88 (3-4), 449–466, 2016.
• [8] P. Danchev and J. Matczuk, n-torsion clean rings, in: Rings, modules and codes, Contemp. Math. 727, Amer. Math. Soc. Providence, RI, 71–82, 2019.
• [9] J. Han and W.K. Nicholson, Extension of clean rings, Comm. Algebra, 29 (6), 2589– 2595, 2001.
• [10] I. Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. 54 (6), 575– 580, 1948.
• [11] M.T. Koşan, The p.p. property of trivial extensions, J. Algebra Appl. 14 (8), 1550124, 5 pp., 2015.
• [12] M.T. Koşan, A. Leroy and J. Matczuk, On UJ-rings, Comm. Algebra, 46 (5), 2297– 2303, 2018.
• [13] T.Y. Lam, A First Course in Noncommutative Rings (second Ed.), Springer Verlag, New York, 2001.
• [14] J. Levitzki, On the structure of algebraic algebras and related rings, Trans. Amer. Math. Soc. 74, 384–409, 1953.
• [15] M. Marianne, Rings of quotients of generalized matrix rings, Comm. Algebra, 15 (10), 1991–2015, 1987.
• [16] W.K. Nicholson, Semiregular modules and rings, Canad. J. Math. 28 (5), 1105–1120, 1976.
• [17] W.K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229, 269-278, 1977.
• [18] W.K. Nicholson, Strongly clean rings and Fittings lemma, Comm. Algebra, 27, 3583– 3592, 1999.
• [19] S. Sahinkaya and T. Yildirim, UJ-endomorphism rings, The Mathematica Journal, 60 (83), 186–198, 2018.