Timeline for The existence of two maximal ideals with the same set of idempotents
Current License: CC BY-SA 4.0
5 events
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| Aug 15, 2021 at 15:38 | comment | added | Keith Kearnes | @Antony: Use Zorn's Lemma. Order the pairs $(A,B)$ which satisfy the conditions by $(A,B)\sqsubseteq (X,Y)$ if $A\subseteq X$ and $B\subseteq Y$. This set is inductively ordered. Any $(A,B)$ satisfying the conditions can be extended to a maximal pair $(A',B')$ satisfying the conditions. Using Lemma 1, one can show that $A'$ and $B'$ have the same idempotents. Using Lemma 2, one can show that for any idempotent $e\in R$, either $e$ is in both $A'$ and $B'$ or $1-e$ is in both $A'$ and $B'$. Then, following Pace's sketch, if $(A',B')$ is a maximal pair, then $A'$ and $B'$ must be maximal ideals. | |
| Aug 15, 2021 at 15:26 | history | edited | Keith Kearnes | CC BY-SA 4.0 |
added 40 characters in body
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| Aug 15, 2021 at 9:14 | comment | added | Antony | @Keith Kearnes: As I commented above, how does your argument work in the second lemma when the set $\{e^2=e\in R\mid e\not\in A\text{ }\text{and} \text{ } 1-e\not\in A\}$ is an infinite set? your Lemma works for finitely many steps! Please explain a little more. | |
| Aug 14, 2021 at 18:05 | comment | added | Martin Brandenburg | Since the OP didn't include any details about the context or his own thoughts (it may even be homework, who knows), less detailed answers are probably preferable. | |
| Aug 14, 2021 at 17:33 | history | answered | Keith Kearnes | CC BY-SA 4.0 |