Definition talk:Prime Ideal of Ring/Commutative and Unitary Ring

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"Can we change the assumption Definition:Commutative and Unitary Ring to Definition:Commutative Ring? The unused unitary assumption makes these definitions less applicable."

My immediate response would be: No, because a specific effort was made by a contributor to define this concept specifically for a commutative and unitary ring. I don't know what that specific reason was, he did not share his unsourced, uncorroborated and somewhat scattergun agenda. I am reluctant to throw that all away because there's no immediate vision of where this was going.
There is already a definition for the general ring. Are you able to just use that? If you do need to use a definition specifically for a general commutative ring (which may or may not be unitary), then feel free to add another subpage to the existing structure, and if it turns out that we really do not need the Commutative and Unitary Ring version, we can subsequently remove it without compromising the validity of anything. --prime mover (talk) 14:39, 1 July 2022 (UTC)
I need the equivalence of Definition1, Definition2 and Definition3. Its proof does not require the unity at all. The same holds for many subsequent results, too. Ultimately, we should remove the with unity-assumption, where it is unnecessary.
Hang on, I'm questioning this. Why do you "need the equivalence of Definition1, Definition2 and Definition3"? --prime mover (talk) 22:17, 1 July 2022 (UTC)
I guess the reason for this unnecessary assumption is simply that many books on commutative algebra (e.g. Atiyah & MacDonald) generally defines the ring as with unity.
I must wait a bit. No idea what to do. --Usagiop (talk) 16:32, 1 July 2022 (UTC)
The best approach to populating this site is to pick a source work and go over it in rigorous detail. At the least, every definition should be documented and compared with existing definitions, and merging as appropriate. It takes considerable critical thinking skills and attention to detail. Unfortunately this can be tedious and time-consuming.
The danger is always that one source work takes a subtly different approach from another, and if you take a result from one work and refer back to definitions sourced from a different work, you can find the differences are significant. Hence our rigorous attention to detail and a rigid approach to the source flow.
But of course you know and understand all this, because you have read the help pages and thoroughly understand our site philosophy. --prime mover (talk) 18:44, 1 July 2022 (UTC)
I checked some other algebra books (Lang, Reid). Unfortunately, they all assume with unity from beginning on. So, we need to find a more generally written reference book or to restrict ourselves to this setting. I would try the first approach because assuming the unnecessary with unity condition everywhere in $\mathsf{Pr} \infty \mathsf{fWiki}$ needlessly limits reusability of the existing results. It is painful to create unessential variations for without unity cases. --Usagiop (talk) 20:03, 1 July 2022 (UTC)
But maybe it is OK to always assume with unity for commutative rings. I will check more sources.--Usagiop (talk) 20:12, 1 July 2022 (UTC)
If everybody considers prime ideals With Unity as a significant object of definition, then there is clearly a reason for it.
I will re-state my answer to the other question, which is equally important: if you don't have a source work which specifically discusses "a prime ideal on a commutative ring" as specifically a significantly "different" object from a prime ideal on a non-commutative ring, which definition we already have, then there's no point adding it.
You're basing your work on a specific source, or a specific set of sources, aren't you? This is all elementary material. It is in a book somewhere. Follow that book. --prime mover (talk) 22:16, 1 July 2022 (UTC)
OK, considering prime ideals for non-unital rings are minor. Even Bourbaki assumes rings are unital. So, I have no objection to just assume it everywhere. Seemingly I was confused with references.--Usagiop (talk) 00:34, 2 July 2022 (UTC)