  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.
