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

DOI:
https://doi.org/10.1090/S0002-9939-10-10252-4

Published electronically:
January 13, 2010

MathSciNet review:
2587445

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a ring and polynomials satisfying , we prove that the ideal generated by products of the coefficients of and 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 is a left -nilpotent ideal, then the ideal formed by the coefficients of polynomials in is also left -nilpotent.

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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

Email:
pace@math.byu.edu

DOI:
https://doi.org/10.1090/S0002-9939-10-10252-4

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