A prime ideal is either maximal right ideal or small right ideal.

Definition:- A right ideal $I$ of a ring $R$ is called small right ideal if $I+J=Rimplies J=R$ for any right ideal $I$ of $R$.

My Question:- A prime ideal is either a maximal right ideal or a small right ideal.

I have tried to find counterexamples but I couldn’t find. So, I have tried to prove it in many ways but I couldn’t do so also. So, I couldn’t conclude that above statement is true or false. I need your suggestion in this problem.

My attempt:- Suppose that $P$ is not a small right ideal then there is a proper right ideal $J$ of $R$ such that $P+J=R$. Now we need to show that $P$ is maximal. Let $Q$ be a right ideal of $R$ such that $Psubseteq Qsubseteq R$.Then we show that either $Q=P$ or $Q=R$. If possible, assume that $Qneq P$ then there is an element $xin Qbackslash P.$