Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-01T15:46:13.332Z Has data issue: false hasContentIssue false
  • English
  • Français

Nilpotent, Algebraic and Quasi-Regular Elements in Rings and Algebras

Part of: Radicals and radical properties of rings Conditions on elements

Published online by Cambridge University Press:  29 December 2016

Nik Stopar
Nik Stopar*
Affiliation:
Faculty of Electrical Engineering, University of Ljubljana, Tržaška cesta 25, 1000 Ljubljana, Slovenia (nik.stopar@fe.uni-lj.si)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that an integral Jacobson radical ring is always nil, which extends a well-known result from algebras over fields to rings. As a consequence we show that if every element x of a ring R is a zero of some polynomial px with integer coefficients, such that px(1) = 1, then R is a nil ring. With these results we are able to give new characterizations of the upper nilradical of a ring and a new class of rings that satisfy the Köthe conjecture: namely, the integral rings.

Keywords

π-algebraic elementnil ringintegral ringquasi-regular elementJacobson radicalupper nilradical

MSC classification

Secondary: 16N40: Nil and nilpotent radicals, sets, ideals, rings 16N20: Jacobson radical, quasimultiplication 16U99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 3 , August 2017 , pp. 753 - 769
Copyright
Copyright © Edinburgh Mathematical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Eisenbud, D., Commutative algebra with a view toward algebraic geometry (Springer, 1995).Google Scholar
2. Gardner, B. J. and Wiegandt, R., Radical theory of rings (Marcel Dekker, New York, 2004).Google Scholar
3. Krempa, J., Logical connections between some open problems concerning nil rings, Fund. Math. 76 (1972), 121–130.CrossRefGoogle Scholar
4. Levitzki, J., A theorem on polynomial identities, Proc. Am. Math. Soc. 1 (1950), 334–341.Google Scholar
5. Smoktunowicz, A., Some open results related to Köthe's conjecture, Serdica Math. J. 27 (2001), 159–170.Google Scholar
6. Smoktunowicz, A., Some results in noncommutative ring theory, in Proc. International Congress of Mathematicians, Madrid, Spain, 2006.Google Scholar
7. Szász, F. A., Radicals of rings (Wiley, 1981).Google Scholar
8. Weng, J. H. and Wu, P. Y., Products of unipotent matrices of index 2, Linear Alg. Applic. 149 (1991), 111–123.Google Scholar
9. Yonghua, X., On the Koethe problem and the nilpotent problem, Sci. Sinica A26 (1983), 901–908.Google Scholar