# Answer to Question #17659 in Abstract Algebra for Tsit Lam

Question #17659

Give an example to show that

rad(R) ⊆ {r ∈ R : r +U(R) ⊆ U(R)}.

need not be an equality.

rad(R) ⊆ {r ∈ R : r +U(R) ⊆ U(R)}.

need not be an equality.

Expert's answer

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.
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