Korean J. Math. Vol. 24 No. 3 (2016) pp.307-317
DOI: https://doi.org/10.11568/kjm.2016.24.3.307

On NI and quasi-NI rings

Main Article Content

Dong Hwa Kim
Seung Ick Lee
Yang Lee
Sang Jo Yun


Let $R$ be a ring. It is well-known that $R$ is {\it NI} if and only if $\sum_{i=0}^nRa_iR$ is a nil ideal of $R$ whenever a polynomial $\sum_{i=0}^na_ix^i$ is nilpotent, where $x$ is an indeterminate over $R$. We consider a condition which is similar to the preceding one:
$\sum_{i=0}^nRa_iR$ contains a nonzero nil ideal of $R$ whenever $\sum_{i=0}^na_ix^i$ over $R$ is nilpotent. A ring will be said to be {\it quasi-NI} if it satisfies this condition. The structure of quasi-NI rings is observed, and various examples are given to situations which raised naturally in the process.

Article Details

Supporting Agencies

This work was supported by 2-year Research Grant of Pusan National University.


[1] G.F. Birkenmeier, J.Y. Kim and J.K. Park, A connection between weak regularity and the simplicity of prime factor rings, Proc. Amer. Math. Soc. 122 (1994), 53–58. Google Scholar

[2] J.L. Dorroh, Concerning adjunctins to algebras, Bull. Amer. Math. Soc. 38 (1932), 85–88. Google Scholar

[3] K.R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979. Google Scholar

[4] D. Handelman and J. Lawrence, Strongly prime rings, Tran. Amer. Math. Soc. 211 (1975), 209–223. Google Scholar

[5] C.Y. Hong and T.K. Kwak, On minimal strongly prime ideals, Comm. Algebra 28 (2000), 4867–4878. Google Scholar

[6] C. Huh, H.K. Kim and Y. Lee, On rings whose strongly prime ideals are completely prime, Comm. Algebra 26 (1998), 595–600. Google Scholar

[7] C. Huh, C.I. Lee and Y. Lee, On rings whose strongly prime ideals aAre completely prime, Algebra Colloq. 17 (2010), 283–294. Google Scholar

[8] S.U. Hwang, Y.C. Jeon and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302 (2006), 186–199. Google Scholar

[9] 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

[10] N.K. Kim, Y. Lee and S.J. Ryu, An ascending chain condition on Wedderburn radicals, Comm. Algebra 34 (2006), 37–50. Google Scholar

[11] J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359–368. Google Scholar

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

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