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

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.

**1.**David E. Fields,*Zero divisors and nilpotent elements in power series rings*, Proc. Amer. Math. Soc.**27**(1971), 427–433. MR**0271100**, 10.1090/S0002-9939-1971-0271100-6**2.**B. J. Gardner,*Some aspects of 𝑇-nilpotence*, Pacific J. Math.**53**(1974), 117–130. MR**0360667****3.**B. J. Gardner and R. Wiegandt,*Radical theory of rings*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 261, Marcel Dekker, Inc., New York, 2004. MR**2015465****4.**Abraham A. Klein,*Rings of bounded index*, Comm. Algebra**12**(1984), no. 1-2, 9–21. MR**732182**, 10.1080/00927878408822986**5.**Abraham A. Klein,*The sum of nil one-sided ideals of bounded index of a ring*, Israel J. Math.**88**(1994), no. 1-3, 25–30. MR**1303489**, 10.1007/BF02937505**6.**T. Y. Lam,*A first course in noncommutative rings*, 2nd ed., Graduate Texts in Mathematics, vol. 131, Springer-Verlag, New York, 2001. MR**1838439****7.**André Leroy and Jerzy Matczuk,*Goldie conditions for Ore extensions over semiprime rings*, Algebr. Represent. Theory**8**(2005), no. 5, 679–688. MR**2189578**, 10.1007/s10468-005-0707-y**8.**E. M. Patterson,*On the radicals of rings of row-finite matrices*, Proc. Roy. Soc. Edinburgh Sect. A**66**(1961/1962), 42–46. MR**0142582****9.**Edmund R. Puczyłowski,*Nil ideals of power series rings*, J. Austral. Math. Soc. Ser. A**34**(1983), no. 3, 287–292. MR**695913****10.**E. R. Puczyłowski and Agata Smoktunowicz,*The nil radical of power series rings*, Israel J. Math.**110**(1999), 317–324. MR**1750432**, 10.1007/BF02808186**11.**N. E. Sexauer and J. E. Warnock,*The radical of the row-finite matrices over an arbitrary ring*, Trans. Amer. Math. Soc.**139**(1969), 287–295. MR**0238889**, 10.1090/S0002-9947-1969-0238889-9**12.**Agata Smoktunowicz,*Amitsur’s conjecture on polynomial rings in 𝑛 commuting indeterminates*, Math. Proc. R. Ir. Acad.**102A**(2002), no. 2, 205–213. MR**1961638**, 10.3318/PRIA.2002.102.2.205**13.**Agata Smoktunowicz and E. R. Puczyłowski,*A polynomial ring that is Jacobson radical and not nil*, Israel J. Math.**124**(2001), 317–325. MR**1856524**, 10.1007/BF02772627**14.**Julius M. Zelmanowitz,*Radical endomorphisms of decomposable modules*, J. Algebra**279**(2004), no. 1, 135–146. MR**2078391**, 10.1016/j.jalgebra.2004.04.004

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
16N40,
16U80

Retrieve articles in all journals with MSC (2000): 16N40, 16U80

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