Let $R$ be an Artinian ring with unity. Suppose that the ideal $I\subseteq R$ and the corresponding quotient $R/I$ are both commutative Artinian principal ideal rings, show that $R$ is a principal ideal ring.
My attempt: Suppose that $I=\mathcal{N}(R)$, that is, $I$ coincides with the nil radical, then $R$ is a principal ideal ring since $\mathcal{N}(R)$ is principal ideal ring and $R$ is Artinian.
The remaining case is if $I\subset \mathcal{N}(R)$, then there exist $I\subset J\subset \mathcal{N}(R)$. This is where I run out of ideas.
This is false. For instance, if $R=\mathbb{F}_2[x,y]/(x^2,xy,y^2)$ and $I=(x)$ then $I$ is a principal ideal rng and $R/I\cong \mathbb{F}_2[y]/(y^2)$ is a principal ideal ring but $R$ is not a principal ideal ring since the ideal $(x,y)$ is not principal.