Definition talk:Generated Ideal of Ring

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I found a definition and proposition of generated ideal and principal ideal.

Although the set $I$ of linear combinations of elements of $S$ is an ideal of $R$, but it is not the definition of generated ideal in this book. In addition, if $R$ is without unity, it is possible to be $S \nsubseteq I$. What do you think of ideal generation? --Fake Proof (talk) 05:46, 25 July 2021 (UTC)

Beware, this page was edited into its current form by an omniscient idiot. There are three different definitions here, but they are not equivalent because they do not apply to the same objects.
Feel free to analyse the differences between what you see on this page (or rather, suite of pages) and what you have in your source work, and see whether you can reconcile those differences. --prime mover (talk) 06:52, 25 July 2021 (UTC)
I agree with Definition 1, but I want to separate it into two equivalent ones. The definitions of generated ideal and principal ideal in the book (Definition $6.2.28$) are:
  • The smallest ideal containing a subset $S$ is called the ideal generated by $S$. The smallest ideal containing a single element $x \in R$ is called the principal ideal generated by $x$.
There is a theorem (Proposition $6.2.27$) for generated ideals.
  • Let $\gen S$ denote the additive subgroup of $R$ generated by $S$. Then $\gen S + R S + S R + R S R$ is the smallest ideal of $R$ containing $S$. (Definitions of $R S, S R, R S R$ are in the book.)
Instead of definitions 2 and 3, it can be another equivalent definition, or a theorem.
Then some theorems such as Principal Ideal is Ideal and Principal Ideal is Smallest Ideal seem to be meaningless with the new definition of principal ideal. --Fake Proof (talk) 08:08, 25 July 2021 (UTC)
Yes I see what you say. But, as I say, this set of pages has a problem that needs to be cleared up first. --prime mover (talk) 10:13, 25 July 2021 (UTC)

Refactoring

In order to prove Ring by Idempotent and that an ideal generated by a single element can be desribed as a coset, I want to clean this page a little bit.

Unfortunately I'm not sure what should be done here, and I don't know what 'reverse out' means.

The three definitions given at the moment are a mix of equivalent definitions in the one- and the two-sided setups.

There are :

  • Generated left ideal as intersection of ideals containing S
  • Generated left ideal in terms of linear combinations of elements of S
  • Generated right ideal as intersection of ideals containing S
  • Generated right ideal in terms of linear combinations of elements of S
  • Generated two-sided ideal as intersection of two-sided ideals containing S
  • Generated two-sided ideal as in terms of left and right linear combinations of elements of S

All six definitions can be given for any ring, indepentent of unity or commutativity.

Then one can prove, that for commutative rings everything is two-sided.

In particular I propose the following names:

--Wandynsky (talk) 16:14, 30 July 2021 (UTC)

A note on the equivalence discussion above:
As you have correctly noticed, one has to be careful for non-unital rings.
For example:
$\gen S + RS$ is Definition:Generated Ideal of Ring/Left Ideal/Definition 2.
$\gen S + RS + SR + RSR$ is Definition:Generated Ideal of Ring/Two-Sided Ideal/Definition 2.
and so on.
In the unital case this reduces to:
$RS$ (left case)
$RSR$ (two-sided case)
Because of the inclusions $\gen S \subset RS, SR \subset RSR$.
This has to be proved separately.
In the commutative case one can prove $RS = SR$ and $RSR = RRS \subset RS$, so it reduces to:
$\gen S + RS$ (left case)
$\gen S + RS$ (two-sided case)
In the commutative unital case $\gen S \subset RS$, so everything is $RS$.
I propose to prove all of this separately and independent of all definitions.

--Wandynsky (talk) 16:29, 30 July 2021 (UTC)

By "reverse out" I mean "remove the latest changes".
It may appear that I have a vendetta against Barto, as much of what I do consists of removing his changes. But in fact, while claiming to be highly knowledgeable, much of his work has been since shown to be flawed. It is usually expected that work be backed up by hard copy sources, but he was reluctant to do so. Hence his additions could not (generally) be corroborated for authenticity, and in many cases have been shown to be incorrect.
However, he was also committed to restructing the website into how he thought it ought to be, which was often against the way it had evolved. (There are reasons the site is structured the way it is -- and as such, when maintained properly in line with the published house style rules -- its internal integrity is preserved.)
This is a case in point. The approach to definitions that present multiple definitions in the manner "Definition 1", "Definition 2", etc. is designed specifically for completely equivalent definitions of the same entity. However, in this case the definitions do not apply to the "same entity": one is for a general ring, one is for a commutative ring with unity, and one is for a ring with unity. And the definitions for each of these are differently structured.
What we would really like to do is to present definitions which can be applied to: a) all rings, b) rings with unity, c) commutative rings with unity, and keep them separate. I suggest that some of the given definitions may apply to all of these, while others may apply to not all of these. We need to present, for each category of ring, all and only those definitions which apply to each separately. Hence we may have Definition:Generated Ideal of Commutative Ring with Unity/Definition 1, ... 2, ...3; Definition:Generated Ideal of Ring with Unity/Definition 1, ... 2; Definition:Generated Ideal of Ring (by implication that there is only one definition).
Equivalence proof for the 3 defs of the first woult then probably invoke the equivalence proof for the 2 defs of the second.
And then you have those other categories of rings, ideals, left ideals, right ideals, and everything associated. It would be excellent if all these different possibilities can be addressed, and presented in the manner according to the house style. On that note, please be aware of our style, which does not endorse direct use of parenthesis markers. The compound constructs \paren, \struct, \tuple, \map, and so on, are to be used exclusively. --prime mover (talk) 18:52, 30 July 2021 (UTC)
What you describe is exactly the way I would go. My above comment describes everything, that applies to general rings. If you like I can start to rebuild the parts (as good as I can, hoping you won't have to clean too much).
Is the naming style Definition:Generated Ideal of Ring/Left Ideal/Definition 1 okay? Or rather Definition:Generated Left Ideal of Ring/Definition 1? I wasn't sure if double subpages are a good idea.
Using the style Definition:Generated Ideal of Ring/Left Ideal/Definition 1 is better. You then get the parent page appearing as a link at the top of the child page. --prime mover (talk) 05:25, 31 July 2021 (UTC)
I noticed that Barto had several erroneous edits. He should be encouraged to check carefully if what he writes is mathematically precise. Also own inventions that don't make too much sense should be avoided (like the polynomial ring thing). What I learned slowly and painfully over time is, that it is highly nontrivial to write down meaningful general definitions and this is the part that should be treated very cleanly in such a big project as the $\mathsf{Pr} \infty \mathsf{fWiki}$. There are usually not many definitions that are good and not already written down somewhere in some form. As you've mentioned somewhere before, there are many authors with very different ideas of how the building of mathematics looks like. Deep changes like his should be carefully discussed. --Wandynsky (talk) 23:54, 30 July 2021 (UTC)
Barto was been blocked from contributing further a few years ago. --prime mover (talk) 05:25, 31 July 2021 (UTC)
Here Sum of All Ring Products is Additive Subgroup the definition of these $RS$ sets is given implicitly. I couldn't find a definition here, there seems to be none. Also there seems to be no good name for this set. --Wandynsky (talk) 00:38, 31 July 2021 (UTC)
There are numerous places where the definitions may not all have been presented in fully explicit style. As we find them, we can edit the pages to improve them.
Definition of Generated Ideal of Ring: I suggest to separate the current Definition 1 into two equivalent definitions, "intersection of all ideals containing $S$" and "the smallest ideal containing $S$", and prove they are equivalent. I'm preparing for what I want in my user page, but I'm not ready to refactor this page.
Okay, so we have three definitions total: "intersection of all ideals containing $S$", "the smallest ideal containing $S$" and "linear combinations of the form ..." in each case. In theory it would be possible to omit the last one. I don't know whether it is better to prove the equivalent condition or to add it as an equivalent definition. --Wandynsky (talk) 13:45, 31 July 2021 (UTC)
Multiple subpage: As you see multiple subpages like Division Theorem/Positive Divisor/Positive Dividend/Existence/Proof 1, it does not seem to be a problem.
Definition of ring subset product: The definition of ring subset product is different from the ordinary subset product, but I could not find it. I want to create a page for it. Another idea is to consider it the additive subgroup generated by the ordinary product and denote $\gen {S T}$, but I've never seen a source using such notation. --Fake Proof (talk) 05:06, 31 July 2021 (UTC)
$\gen {S T}$ looks very logical. But usually they write $S T$. In ideal theory it would be a burden to always have to write $\gen {I^2}$ instead of $I^2$ ... So I like $ST$ better in the long run. A link to the definition has to be added in any case. --Wandynsky (talk) 13:45, 31 July 2021 (UTC)
I found some probably related pages. Intersection of Ring Ideals Containing Subset is Smallest can be helpful to prove "intersection of all ideals containing $S$", "the smallest ideal containing $S$" are equal. Definition:Product of Ideals of Ring/Definition 1 can be used to define the product. However, this definition is only about product of ideals of a commutative ring, so I want to generalize it. --Fake Proof (talk) 10:51, 3 August 2021 (UTC)
Definition:Product of Ideals of Ring seems to be not referenced by other pages. So it seems safe to generalize the definition to apply to every two ideals of every ring. Then Definition:Product of Subsets of Ring makes sense to be on a different page. You can use Definition:Product of Subsets of Ring in the definition of Definition:Product of Ideals of Ring. When everything is arranged properly I can fill in the proofs. --Wandynsky (talk) 11:48, 3 August 2021 (UTC)
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