Let $R$ be a ring. Then $J(R/J(R))=(0)$ .

Proof:
We will only prove this in the case where $R$ is a unital ring (although it is true without this assumption).

Let $[u] \in J(R/J(R))$ . By one of the characterizations of the Jacobson radical, $1-[r][u]$ is left invertible for all $r \in R$ , so there exists $v \in R$ such that $[v](1-[r][u])=1$ .

Then $v(1-ru)=1-a$ for some $a \in J(R)$ . There is a $w \in R$ such that $w(1-a)=1$ , and we have $wv(1-ru)=1$ .

Since this holds for all $r \in R$ , it follows that $u \in J(R)$ , and therefore $[u]=0$ .