Nilpotent, algebraic and quasi-regular elements in rings and algebras.

Year of edition2015

 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 p_x with integer coefficients, such that p_x(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 K\"othe conjecture, namely the integral rings.

 18 pages

Reference on publication
Stopar N.   Nilpotent, algebraic and quasi-regular elements in rings and algebras. - : , 2015. //, 2015.
1.D. Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, New York, 1995
2.B.J. Gardner, R. Wiegandt: Radical Theory of Rings, Marcel Dekker, New York, 2004
3.J. Krempa: Logical connections between some open problems concerning nil rings, Fund. Math. 76 (1972), 121-130
4.J. Levitzki: A theorem on polynomial identities, Proc. Amer. Math. Soc. 1 (1950), 334-341
5.A.D. Sands: Radicals and Morita contexts, J. Algebra 24 (1973), 335-345
6.A. Smoktunowicz: On primitive ideals in polynomial rings over nil rings, Algebr. Represent. Theory 8 (2005), 69-73
7.A. Smoktunowicz: Some open results related to K?othe's conjecture, Serdica Math. J. 27 (2001), 159-170
8.A. Smoktunowicz: Some results in noncommutative ring theory, Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006
9.F.A. Sz?asz: Radicals of rings, John Wiley & Sons, Chichester, 1981
10.J.H. Weng, P.Y. Wu: Products of unipotent matrices of index 2, Linear Algebra Appl. 149 (1991), 111-123
11.X. Yonghua: On the Koethe problem and the nilpotent problem, Sci. Sin., Ser. A 26 (1983), 901-908

This publication on other resources


Other publications of these authors