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Nilpotent ideals in polynomial and power series rings

Authors: Victor Camillo, Chan Yong Hong, Nam Kyun Kim, Yang Lee and Pace P. Nielsen
Journal: Proc. Amer. Math. Soc. 138 (2010), 1607-1619
MSC (2000): Primary 16N40; Secondary 16U80
Published electronically: January 13, 2010
MathSciNet review: 2587445
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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.

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Victor Camillo
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242

Chan Yong Hong
Affiliation: Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 131-701, Republic of Korea

Nam Kyun Kim
Affiliation: College of Liberal Arts, Hanbat National University, Daejon 305-719, Republic of Korea

Yang Lee
Affiliation: Department of Mathematics Education, Pusan National University, Pusan 609-735, Republic of Korea

Pace P. Nielsen
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

Keywords: Bounded index of nilpotence, left $T$-nilpotent, locally nilpotent, nil, nilpotent, polynomial ring, power series ring
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
Article copyright: © Copyright 2010 By the authors

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