$1+x$ is a unit, then $x \in Rad(A)$?

Question

If we denotes the Jacobson radical of a ring $A$ as $Rad(A)$, then there is a statement as follows:

If ideal $I$ satisfies that $1+x$ is a unit for each $x$ in $I$, then $I\subseteq Rad(A)$.

I fail to proof the statement, and I think the condition may not be enough for the conclusion. Hope someone could help. Thanks!

Answer 1

Hint:

Suppose the coset $1+I$ is all units and that $I\nsubseteq T$ for some maximal right ideal $T$.

Then $I+T=R$, and in particular $-i+t=1$ for some $t\in T$, $i\in I$.

Do you see the contradiction?