Definition talk:Generated Ideal of Ring

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


 * : Chapter $6$. Rings: $\S 2$. Homomorphisms and Ideals

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)
 * Definition:Generated Ideal of Ring/Left Ideal/Definition 1
 * Definition:Generated Ideal of Ring/Left Ideal/Definition 2
 * Definition:Generated Ideal of Ring/Right Ideal/Definition 1
 * Definition:Generated Ideal of Ring/Right Ideal/Definition 2
 * Definition:Generated Ideal of Ring/Two-Sided Ideal/Definition 1
 * Definition:Generated Ideal of Ring/Two-Sided Ideal/Definition 2


 * 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)