Let $R$ be a commutative Artinian ring all of whose maximal ideals are principal. I wish to show that in fact all ideals are $R$ are principal.
As is outlined here Every maximal ideal is principal. Is $R$ principal?, a proof by Kaplansky gives this for Noetherian rings. This immediately answers my question.
Could we answer the question instead this way: By first showing that each local Artinian ring has this property, and then noting that each Artinian ring can be written as a product of Artinian local rings? These are Proposition 8.8 and Theorem 8.7 in Atiyah-Macdonald, respectively.
Suppose $R \cong R_1 \times \cdots \times R_n$ where $R_i$ are Artinian local rings. Then an ideal $I \subset R$ is of the form $I_1 \times \cdots \times I_n \subset R_1 \times \cdots \times R_n$ for some ideals $I_i \subset R_i$. By above, each $I_i$ is principal. So $I_1 \times \cdots \times I_n$ is principal $ \implies I $ is principal.
Is this correct or did I do a dumb thing?