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

TL;DR

This paper proves that integral Jacobson radical rings are nil and characterizes rings where each element satisfies a polynomial with specific properties, extending known results and providing new insights into ring structure.

Contribution

It extends the nilpotency result from algebras over fields to integral Jacobson radical rings and characterizes rings with polynomial identities, including integral rings satisfying the K"othe conjecture.

Findings

Integral Jacobson radical rings are always nil.

Rings where each element is a root of a polynomial p_x with p_x(1)=1 are nil.

New characterizations of the upper nilradical and classes of rings satisfying the K"othe conjecture.

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 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.