Let $R$ be an Artinian ring with identity and let $I$ be an ideal of $R$. Prove that $I$ and $R/I$ are principal ideal rings if and only if $R$ is a principal ideal ring.
Proof: ($\Leftarrow$) If $R$ is a PIR and artinian, then it is precisely a $0$-dimensional Noetherian ring. But for any such ring, there is a canonical direct product decomposition $$R\cong \prod R_m$$ where $m$ varies through the finitely many maximal ideals of $R$. Thus, $R$ being PIR, its Jacobson radical $J(R)$ must be a principal ideal.
($\Rightarrow$) However, I have failed to prove the this side. What I am sure of is that if $I$ is equal to the radical of $R$, then $R$ is a PIR iff its radical is a PIR.