When each prime ideal is maximal [duplicate]
Solution 1:
Yes, what you are talking about is true. The rings you start off describing are called von Neumann regular rings. In fact:
For a commutative ring $R$, all prime ideals of $R$ are maximal iff the Jacobson radical $J(R)$ of $R$ is nil and $R/J(R)$ is von Neumann regular.
Since $J(R)=Nil(R)$ in this case, requiring that $R$ is reduced implies that $J(R)=\{0\}$, so you get that a commutative reduced ring for which all prime ideals are maximal is von Neumann regular.