Equivalent condition for Jacobson radical Matsumura, Commutative Ring Theory, page 3, asks this: 

If $x \in A$ has the property that $1 + Ax$ consists entirely of units, then $x \in \operatorname{rad}(A)$. Prove this.

How to show this?

Matsumura says that $\operatorname{rad}(A)$ is the intersection of all maximal ideals of $A$. If $x\notin \operatorname{rad}(A)$, then there is a maximal ideal $m$ such that $x\notin m$, so $m+Ax=A$. Can you deduce the result now?
