Nilpotent ideals in polynomial and power series rings
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- by Victor Camillo, Chan Yong Hong, Nam Kyun Kim, Yang Lee and Pace P. Nielsen PDF
- Proc. Amer. Math. Soc. 138 (2010), 1607-1619
Abstract:
Given a ring $R$ and polynomials $f(x),g(x)\in R[x]$ satisfying $f(x)Rg(x)=0$, we prove that the ideal generated by products of the coefficients of $f(x)$ and $g(x)$ is nilpotent. This result is generalized, and many well known facts, along with new ones, concerning nilpotent polynomials and power series are obtained. We also classify which of the standard nilpotence properties on ideals pass to polynomial rings or from ideals in polynomial rings to ideals of coefficients in base rings. In particular, we prove that if $I\leq R[x]$ is a left $T$-nilpotent ideal, then the ideal formed by the coefficients of polynomials in $I$ is also left $T$-nilpotent.References
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Additional Information
- Victor Camillo
- Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
- Email: camillo@math.uiowa.edu
- Chan Yong Hong
- Affiliation: Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 131-701, Republic of Korea
- Email: hcy@khu.ac.kr
- Nam Kyun Kim
- Affiliation: College of Liberal Arts, Hanbat National University, Daejon 305-719, Republic of Korea
- Email: nkkim@hanbat.ac.kr
- Yang Lee
- Affiliation: Department of Mathematics Education, Pusan National University, Pusan 609-735, Republic of Korea
- Email: ylee@pusan.ac.kr
- Pace P. Nielsen
- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- MR Author ID: 709329
- Email: pace@math.byu.edu
- Received by editor(s): June 9, 2009
- Received by editor(s) in revised form: September 15, 2009
- Published electronically: January 13, 2010
- Communicated by: Birge Huisgen-Zimmermann
- © Copyright 2010 By the authors
- Journal: Proc. Amer. Math. Soc. 138 (2010), 1607-1619
- MSC (2000): Primary 16N40; Secondary 16U80
- DOI: https://doi.org/10.1090/S0002-9939-10-10252-4
- MathSciNet review: 2587445