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upper nilradical

Proceedings of the Edinburgh Mathematical Society (1)

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1 results

Nilpotent, Algebraic and Quasi-Regular Elements in Rings and Algebras
Part of
- Radicals and radical properties of rings
- Conditions on elements
Nik Stopar
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 60 / Issue 3 / August 2017

Published online by Cambridge University Press:

29 December 2016, pp. 753-769
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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.