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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 ([email protected])
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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 

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