An equality between right ideals and Jacobson radical

Question

Let $R$ be a unital ring with Jacobson radical $J(R)$. If for any right ideal $I$ of $R$ we have $I+J(R)=I^2+J(R)$ could we deduce that $I$ is the sum of an idempotent right ideal and a nil right ideal? (By the hypothesis, of course $I\subseteq I^2+J(R)$.)

Answer 1

The answer is no, a counterexample follows.

Let $R$ be any DVR with unique maximal ideal $J$ (which is also the Jacobson radical of $R$). In particular the equality $$I+J=J=I^2+J$$ is trivially satisfied for all ideals $I$. Moreover, recall that all ideals have the form $J^n$ for some $n \ge 0$. The unque idempotent ideal is thus $J^0=R$, and the unique nil ideal is $0$ (there are no nonzero nilpotent elements). Hence the sum of an idempotent ideal and a nil ideal is necessarily $R$, which turns out to be distinct from $I$ in general.