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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?

Pentaki
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1 Answers1

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This is a valid reduction of the problem from commutative Artinian rings to commutative local Artinian rings.

Now... have you managed to do it in the local case yet?

Here is a different proof. Suppose there exists a nonprincipal ideal. Then there is a maximal nonprincipal ideal, which is prime. But prime ideals are maximal in Artinian rings, so it must be principal by our initial assumption, and that is a contradiction.

rschwieb
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  • Oh yes, I like your approach better. It takes some thought to show that each maximal ideal in $R_i$ is principal. Thoughts I don't want to think – Pentaki Sep 27 '18 at 18:29