Korean J. Math.  Vol 27, No 1 (2019)  pp.141-150
DOI: https://doi.org/10.11568/kjm.2019.27.1.141

On a ring property related to nilradicals

Hai-lan Jin, Zhelin Piao, Sang Jo Yun


In this article we investigate the structure of rings in which lower nilradicals coincide with upper nilradicals. Such rings shall be said to be quasi-2-primal. It is shown first that the K$\ddot{\text{o}}$the's conjecture holds for quasi-2-primal rings. So the results in this article may provide interesting and useful information to the study of nilradicals in various situations. In the procedure we study the structure of quasi-2-primal rings, and observe various kinds of quasi-2-primal rings which do roles in ring theory.


equal-nilradical ring, lower nilradical, upper nilradical, Jacobson radical, polynomial ring, matrix ring

Subject classification

16N40, 16N20, 16S36, 16S50


Full Text:



D.D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), 2265–2272. (Google Scholar)

R. Antoine, Nilpotent elements and Armendariz rings, J. Algebra 319 (2008), 3128–3140. (Google Scholar)

E.P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470–473. (Google Scholar)

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 (1993), 102–129. (Google Scholar)

G.F. Birkenmeier, J.Y. Kim, and J.K. Park, Regularity conditions and the simplicity of prime factor rings, J. Pure Appl. Algebra 115 (1997), 213–230. (Google Scholar)

V. Camillo and P.P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra 212 (2008), 599–615. (Google Scholar)

V. Camillo, C.Y. Hong, N.K. Kim, Y. Lee, and P.P. Nielsen, Nilpotent ideals in polynomial and power series rings, Proc. Amer. Math. Soc. 138 (2010), 1607– 1619. (Google Scholar)

K.R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979. (Google Scholar)

S.U. Hwang, Y.C, Jeon, and Y. Lee, Structure and topological conditions of NI (Google Scholar)

rings, J. Algebra 302 (2006), 186–199. (Google Scholar)

Y.C. Jeon, H.K. Kim, Y. Lee, and J.S. Yoon, On weak Armendariz rings, Bull. Korean Math. Soc. 46 (2009), 135–146. (Google Scholar)

N.K. Kim, K.H. Kim, and Y. Lee, Power series rings satisfying a zero divisor property, Comm. Algebra 34 (2006), 2205–2218. (Google Scholar)

N.K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), 477–488. (Google Scholar)

G. K ̈othe, Die Struktur der Ringe deren Restklassenring nach dem Radikal voll- standing irreduzibel ist, Math. Z. 32 (1930), 161–186. (Google Scholar)

J. Krempa, Logical connections among some open problems concerning nil rings, Fund. Math. 76 (1972), 121–130. (Google Scholar)

T.Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991. (Google Scholar)

G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (2001), 2113–2123. (Google Scholar)

G. Marks, A taxonomy of 2-primal rings, J. Algebra 266 (2003), 494–520. (Google Scholar)

J.V. Neumann, On regular rings, Proceedngs of the National Academy of Sciences, 22 (1936), 707–713. (Google Scholar)

M.B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), 14–17. (Google Scholar)

L.H. Rowen, Ring Theory, Academic Press, San Diego, 1991. (Google Scholar)

A.D. Sands, Radical and Morita contex, J. Algebra 24 (1973), 335–345. (Google Scholar)


  • There are currently no refbacks.

ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr