NILRADICALS OF POWER SERIES RINGS AND NIL POWER SERIES RINGS

- Journal title : Journal of the Korean Mathematical Society
- Volume 42, Issue 5, 2005, pp.1003-1015
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2005.42.5.1003

Title & Authors

NILRADICALS OF POWER SERIES RINGS AND NIL POWER SERIES RINGS

HUH, CHAN; KIM, CHOL ON; KIM, EUN JEONG; KIM, HONG KEE; LEE, YANG;

HUH, CHAN; KIM, CHOL ON; KIM, EUN JEONG; KIM, HONG KEE; LEE, YANG;

Abstract

Klein proved that polynomial rings over nil rings of bounded index are also nil of bounded index; while Puczylowski and Smoktunowicz described the nilradical of a power series ring with an indeterminate. We extend these results to those with any set of commuting indeterminates. We also study prime radicals of power series rings over some class of rings containing the case of bounded index, finding some examples which elaborate our arguments; and we prove that R is a PI ring of bounded index then the power series ring R[[X]], with X any set of indeterminates over R, is also a PI ring of bounded index, obtaining the Klein's result for polynomial rings as a corollary.

Keywords

nilradical;Wedderburn radical;polynomial ring;power series ring;nil ring of bounded index;

Language

English

Cited by

1.

2.

References

1.

S. A. Amitsur, Nil radicals. Historical notes and some new results, Associative Rings, Modules, and Radicals (Proc. Colloquium at Keszthely 1971, edited by A. Kertesz) (1973), Janos Bolyai Mathematical Society and North-Holland Publishing Company, Amsterdam, London, and Budapest, 47-65

2.

C. Huh, H. K. Kim, D. S. Lee, Y. Lee, Prime radicals of formal power series rings, Bull. Korean Math. Soc. 38 (2001), 623-633

4.

N. Jacobson, Structure of Rings, vol. 37, Amer. Math. Soc. Colloq. Publ., 1964

6.

A. A. Klein, The sum of nil one-sided ideals of bounded index of a ring, Israel J. Math. 88 (1994), 25-30

7.

E. R. Puczylowski, On radicals of polynomial rings, power series rings and tensor products, Comm. Algebra 8 (1980), 1699-1709

8.

E. R. Puczylowski and A. Smoktunowicz, The nil radical of power series rings, Israel J. Math. 110 (1999), 317-324

9.

L. H. Rowen, Ring Theory, Academic Press, Inc., 1991