×

Nilpotent ideals in polynomial and power series rings. (English) Zbl 1209.16014

The authors start with the following result: Let \(R\) be a ring. If polynomials \(f(x),g(x)\in R[x]\) satisfy \(f(x)Rg(x)=0\) then the ideal generated by products of the coefficients of \(f(x)\) and \(g(x)\) is nilpotent. This result is then used to prove the following interesting theorem: Let \(R\) be a ring. If \(I\) is a left \(T\)-nilpotent ideal in the polynomial ring \(R[x]\), then the ideal formed by the coefficients of polynomials in \(I\) is also left \(T\)-nilpotent.
Several other results concerning nil ideals, Jacobson radical ideals and \(T\)-nilpotent ideals in polynomial rings and power series rings are also obtained.

MSC:

16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16S36 Ordinary and skew polynomial rings and semigroup rings
16U80 Generalizations of commutativity (associative rings and algebras)
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] David E. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc. 27 (1971), 427 – 433. · Zbl 0219.13023
[2] B. J. Gardner, Some aspects of \?-nilpotence, Pacific J. Math. 53 (1974), 117 – 130. · Zbl 0253.16009
[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. · Zbl 1034.16025
[4] Abraham A. Klein, Rings of bounded index, Comm. Algebra 12 (1984), no. 1-2, 9 – 21. · Zbl 0535.16006
[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. · Zbl 0834.16016
[6] T. Y. Lam, A first course in noncommutative rings, 2nd ed., Graduate Texts in Mathematics, vol. 131, Springer-Verlag, New York, 2001. · Zbl 0980.16001
[7] André Leroy and Jerzy Matczuk, Goldie conditions for Ore extensions over semiprime rings, Algebr. Represent. Theory 8 (2005), no. 5, 679 – 688. · Zbl 1090.16011
[8] E. M. Patterson, On the radicals of rings of row-finite matrices, Proc. Roy. Soc. Edinburgh Sect. A 66 (1961/1962), 42 – 46. · Zbl 0145.27103
[9] Edmund R. Puczyłowski, Nil ideals of power series rings, J. Austral. Math. Soc. Ser. A 34 (1983), no. 3, 287 – 292. · Zbl 0522.16005
[10] E. R. Puczyłowski and Agata Smoktunowicz, The nil radical of power series rings, Israel J. Math. 110 (1999), 317 – 324. · Zbl 0934.16016
[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. · Zbl 0191.31803
[12] Agata Smoktunowicz, Amitsur’s conjecture on polynomial rings in \? commuting indeterminates, Math. Proc. R. Ir. Acad. 102A (2002), no. 2, 205 – 213. · Zbl 1030.16013
[13] Agata Smoktunowicz and E. R. Puczyłowski, A polynomial ring that is Jacobson radical and not nil, Israel J. Math. 124 (2001), 317 – 325. · Zbl 1036.16020
[14] Julius M. Zelmanowitz, Radical endomorphisms of decomposable modules, J. Algebra 279 (2004), no. 1, 135 – 146. · Zbl 1109.16027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.