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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeJun 23rd 2014
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexThere was a note at the top of [[ideal]]: >This entry discusseds the notion of _ideal_ in fair generality. For an entry closer to the standard notion see at _[[ideal in a monoid]]_. I\'ve removed this, as it seems exactly backwards. The pages are equally *standard*, but the most *common* notion of ideal is that of an ideal in a ring, and that is the first thing discussed at [[ideal]], and in very basic terms; but at [[ideal in a monoid]], this is discussed only via rings\' being monoids in $Ab$, and it\'s not spelt out.

   There was a note at the top of ideal:

   This entry discusseds the notion of ideal in fair generality. For an entry closer to the standard notion see at ideal in a monoid.

   I've removed this, as it seems exactly backwards. The pages are equally standard, but the most common notion of ideal is that of an ideal in a ring, and that is the first thing discussed at ideal, and in very basic terms; but at ideal in a monoid, this is discussed only via rings' being monoids in $Ab$, and it's not spelt out.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 23rd 2014
   • (edited Jun 23rd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSorry, I suppose that was me. I remember linking to _[[ideal]]_ for the standard notion, then briefly checking what that takes the reader to, and getting away with the impression that it sends the reader not so much to the standard story as to some generalization of it. I realize that I was wrong, and thanks for fixing it. But I do think the Idea-sectionat _[[ideal]]_ could be improved. Let it say what actually the basic idea of ideals is, and don't let it end in such a way (as currently) that makes the impression that not the standard story but some generalization is being discussed. (Of course I can try to edit myself, but will be offline now with no telling when I'll be back.)

   Sorry, I suppose that was me.

   I remember linking to ideal for the standard notion, then briefly checking what that takes the reader to, and getting away with the impression that it sends the reader not so much to the standard story as to some generalization of it.

   I realize that I was wrong, and thanks for fixing it. But I do think the Idea-sectionat ideal could be improved. Let it say what actually the basic idea of ideals is, and don’t let it end in such a way (as currently) that makes the impression that not the standard story but some generalization is being discussed.

   (Of course I can try to edit myself, but will be offline now with no telling when I’ll be back.)

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeJun 24th 2014
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexActually, [[ideal]] doesn\'t have an Idea section; it starts with a Definition section. I tried to make it look more friendly, but perhaps you can write more later.

   Actually, ideal doesn't have an Idea section; it starts with a Definition section. I tried to make it look more friendly, but perhaps you can write more later.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 24th 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI added some further remarks to [[ideal]].

   I added some further remarks to ideal.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 2nd 2016
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI've also been adding stuff to [[ideal in a monoid]]. I quickly whipped up a general context in which ideals in a monoid form a [[quantale]] under the usual product operation on ideals. It might be a little bit ad hoc, so if you know of something nicer... Some of you may be following what I'm up to here. Basically I wanted a nice general context for [[prime ideal theorem]] which would handle all the usual cases (including rings/rigs, possibly noncommutative) in one fell swoop. The plan is to define a prime ideal, in the generality of the context referred to above, as a prime element in a quantale of ideals. Then existence of a prime ideal, for any notion of ideal fitting in that general context, could be deduced from the ultrafilter lemma once we know that the quantale of ideals is compact, according to this [paper](http://dml.cz/bitstream/handle/10338.dmlcz/701805/ActaCarolinae_030-1989-2_20.pdf) which builds on work of Banaschewski. This I find a fairly clean and conceptual approach to prime ideal theorems.

   I’ve also been adding stuff to ideal in a monoid. I quickly whipped up a general context in which ideals in a monoid form a quantale under the usual product operation on ideals. It might be a little bit ad hoc, so if you know of something nicer…

   Some of you may be following what I’m up to here. Basically I wanted a nice general context for prime ideal theorem which would handle all the usual cases (including rings/rigs, possibly noncommutative) in one fell swoop. The plan is to define a prime ideal, in the generality of the context referred to above, as a prime element in a quantale of ideals. Then existence of a prime ideal, for any notion of ideal fitting in that general context, could be deduced from the ultrafilter lemma once we know that the quantale of ideals is compact, according to this paper which builds on work of Banaschewski. This I find a fairly clean and conceptual approach to prime ideal theorems.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 2nd 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexVery interesting! Is it necessary to pass to the bicategory? Or could we just look at $\mathbf{C}/A$ with the monoidal structure induced by the monoid structure of $A$? Also, I notice that you're using $A$ for the monoid, whereas the Definition section of the page uses $S$ for the monoid and $A$ for the ideal.

   Very interesting! Is it necessary to pass to the bicategory? Or could we just look at $\mathbf{C}/A$ with the monoidal structure induced by the monoid structure of $A$?

   Also, I notice that you’re using $A$ for the monoid, whereas the Definition section of the page uses $S$ for the monoid and $A$ for the ideal.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 2nd 2016
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI'll have to think more about your first sentence, but I did want to consider two-sided ideals, and so I thought I wanted a context of two-sided bimodules to begin with. But maybe it's okay under your suggestion (which would be simpler). Yes, thanks for pointing out the clash in notation. I had begun the edits a couple of days ago, attending to the notation originally set out by Toby, and then forgot about it! :-)

   I’ll have to think more about your first sentence, but I did want to consider two-sided ideals, and so I thought I wanted a context of two-sided bimodules to begin with. But maybe it’s okay under your suggestion (which would be simpler).

   Yes, thanks for pointing out the clash in notation. I had begun the edits a couple of days ago, attending to the notation originally set out by Toby, and then forgot about it! :-)

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 3rd 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexMy thought was that the bimodule tensor product over $A$ is a quotient of the ordinary tensor product (in $\mathbf{C}$), so if we're eventually just taking the image in $A$ in both cases, it shouldn't matter.

   My thought was that the bimodule tensor product over $A$ is a quotient of the ordinary tensor product (in $\mathbf{C}$), so if we’re eventually just taking the image in $A$ in both cases, it shouldn’t matter.

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 3rd 2016
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexBut the poset of subobjects of $A$ as an object of $\mathbf{C}$ is different from the poset of subobjects of $A$ as an object of $Mod_A$. Am I missing something?

   But the poset of subobjects of $A$ as an object of $\mathbf{C}$ is different from the poset of subobjects of $A$ as an object of $Mod_A$. Am I missing something?