Skip to main content Accessibility help
×
Home

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

  • Nik Stopar (a1)

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.

Copyright

References

Hide All
1. Eisenbud, D., Commutative algebra with a view toward algebraic geometry (Springer, 1995).
2. Gardner, B. J. and Wiegandt, R., Radical theory of rings (Marcel Dekker, New York, 2004).
3. Krempa, J., Logical connections between some open problems concerning nil rings, Fund. Math. 76 (1972), 121130.
4. Levitzki, J., A theorem on polynomial identities, Proc. Am. Math. Soc. 1 (1950), 334341.
5. Smoktunowicz, A., Some open results related to Köthe's conjecture, Serdica Math. J. 27 (2001), 159170.
6. Smoktunowicz, A., Some results in noncommutative ring theory, in Proc. International Congress of Mathematicians, Madrid, Spain, 2006.
7. Szász, F. A., Radicals of rings (Wiley, 1981).
8. Weng, J. H. and Wu, P. Y., Products of unipotent matrices of index 2, Linear Alg. Applic. 149 (1991), 111123.
9. Yonghua, X., On the Koethe problem and the nilpotent problem, Sci. Sinica A26 (1983), 901908.

Keywords

MSC classification

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

  • Nik Stopar (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed