Question

Give an example to show that
rad(R) ⊆ {r ∈ R : r +U(R) ⊆ U(R)}.

need not be an equality.

Answer 1

Let I(R) be the set onthe RHS, and consider the case R =A[t] where A isa commutative domain. It is easy to see that rad(R) = 0. However, I(R)contains rad(A), since a ∈ rad(A) implies that a+U(R)= a+U(A) ⊆ U(A) = U(R). Thus,we see that equality does not hold for R = A[t] if wechoose A to be any commutative domain that is not J-semisimple.