Talk:Direct Product iff Nontrivial Idempotent

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This could be made into a TFAE with a third statement: Spec(R) is disconnected. --Z423x5c6 (talk) 05:02, 8 June 2021 (UTC)

We try to make pages as simple and self-contained as possible here. Better to put any info concerning $\Spec R$ into a separate page.
TFAE is at its best when presenting an equivalence proof of equivalent definitions -- it standardises the boilerplate. --prime mover (talk) 05:18, 8 June 2021 (UTC)
I see... --Z423x5c6 (talk) 06:08, 8 June 2021 (UTC)

For the 'Explain', this statement appears as an exercise in Atiyah-Macdonald, but they do not use the word non-trivial. The 'non-trivial' decomposition should mean that neither of the compenent is the zero ring (ring with only 1 element) and non-trivial idempotent means idempotent other than 0 and 1. --Z423x5c6 (talk) 06:08, 8 June 2021 (UTC)

If you don't have the literature to hand which supports the material being posted, then probably best to leave well alone.
It's suboptimal to leave a definition on a theorem page. Far better to put it on its own definition page. But if the only source works to hand don't actually provide such a definition, we had best wait till someone turns up who does have access to such source works.
The best of all worlds is when the person who originally writes a $\mathsf{Pr} \infty \mathsf{fWiki}$ page does it by directly referring to a specific source work which they have at their elbow, so to speak, ensuring that all definitions are on said definition pages, and all pages refer back to that source work in the Sources section. This particular page (and a lot of the work on commutative algebra) was not created with that in mind, as it dates from before that strategy began to be implemented. The best strategy for this would be either a) to find the original source work used for this area (it might have been Grillet, I'm not sure), or b) follow through the same area of mathematics from another source work which develops the theory in the same way (preferably both). If you have the patience to work through such a source work yourself (e.g. the Atiyah-Macdonald) then you are welcome to attempt b).
We have found that dipping into books, websites, etc. and cherry-picking the occasional interesting result to put onto $\mathsf{Pr} \infty \mathsf{fWiki}$ is a suboptimal way of working. --prime mover (talk) 06:24, 8 June 2021 (UTC)