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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeSep 12th 2013
   • PermaLink
   Author: zskoda
   Format: MarkdownItex[[essential ideal]]

   essential ideal

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 12th 2013
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThanks. To be linked to [[injective hull]] at some point.

   Thanks. To be linked to injective hull at some point.

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeSep 13th 2013
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   Author: TobyBartels
   Format: MarkdownItexAlready, [[injective hull]] links to [[essential embedding]], which redirects to [[essential ideal]], but the meaning is different (although I\'m sure related).

   Already, injective hull links to essential embedding, which redirects to essential ideal, but the meaning is different (although I'm sure related).

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 13th 2013
   • (edited Sep 13th 2013)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexWell, the idea of that redirect was to connect it to "essential extension" which was defined in [[essential ideal]]. (In a parenthetical aside, "essential embedding" was given as a synonym.) Do you think that's confusing?

   Well, the idea of that redirect was to connect it to “essential extension” which was defined in essential ideal. (In a parenthetical aside, “essential embedding” was given as a synonym.) Do you think that’s confusing?

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeSep 13th 2013
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   Author: TobyBartels
   Format: MarkdownItexI see the *connection*, vaguely, but there is no definition of ‘essential embedding’ at [[essential ideal]], and seeing the redirect makes me think that the inclusion map of submodules is the referent of ‘essential embedding’. Yet, that is not the referent at [[injective hull]], so yes, it is confusing. In particular, at [[injective hull]], ‘essential embedding’ seems to be something with a meaning in a concrete category, while the concepts at [[essential ideal]] apply to abelian categories. Even in a concrete abelian category, I doubt that these agree.

   I see the connection, vaguely, but there is no definition of ‘essential embedding’ at essential ideal, and seeing the redirect makes me think that the inclusion map of submodules is the referent of ‘essential embedding’. Yet, that is not the referent at injective hull, so yes, it is confusing.

   In particular, at injective hull, ‘essential embedding’ seems to be something with a meaning in a concrete category, while the concepts at essential ideal apply to abelian categories. Even in a concrete abelian category, I doubt that these agree.

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 13th 2013
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexOh, jeez. Yes; sorry. I can fix this a little later (but feel free to fix yourself now if you'd like).

   Oh, jeez. Yes; sorry. I can fix this a little later (but feel free to fix yourself now if you’d like).

7. • CommentRowNumber7.
   • CommentAuthorTobyBartels
   • CommentTimeSep 14th 2013
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI don\'t actually know how to fix this, (unless the redirect from [[essential embedding]] to [[essential ideal]] is just wrong, in which case I can undo that).

   I don't actually know how to fix this, (unless the redirect from essential embedding to essential ideal is just wrong, in which case I can undo that).

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeJun 28th 2024
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   Author: zskoda
   Format: MarkdownItexSlightly updated including adding a couple of references and the definition and a redirect for [[uniform module]]. I think it is natural to have it together, unless the entry gets substantially expanded.

   Slightly updated including adding a couple of references and the definition and a redirect for uniform module. I think it is natural to have it together, unless the entry gets substantially expanded.

9. • CommentRowNumber9.
   • CommentAuthorzskoda
   • CommentTimeJun 28th 2024
   • PermaLink
   Author: zskoda
   Format: MarkdownItexOh, no, I looked for it and missed it -- we have a uniform module from before :) (I searched for uniform submodule that is why).

   Oh, no, I looked for it and missed it – we have a uniform module from before :) (I searched for uniform submodule that is why).

10. • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeJul 4th 2024
    • (edited Jul 5th 2024)
    • PermaLink
    Author: zskoda
    Format: MarkdownItexListed Rowen's book, few other minor changes and new properties section. In particular, produced the following reasoning: A monomorphism $h$ is essential iff $g\circ h$ is monic only if $g$ is monic. Indeed, if $h:M\to N$ is essential and $g$ is not monic, then $Ker g\cap Im h\neq 0$ hence $g(Im h)\neq 0$ and $g\circ h$ is not monic. Conversely, suppose $g\circ h$ is monic implies $g$ is monic. If $h$ were not essential then there would be $0\neq K\subset N$ such that $K\cap M = 0$; in that case the cokernel map $g: N\to N/K$ is not monic while $g\circ h$ is monic because $Ker(g)\cap Im(h) = K\cap M = 0$ and $h$ is monic. This is a contradiction. <a href="https://ncatlab.org/nlab/revision/diff/essential+ideal/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/essential+ideal/6">v6</a>, <a href="https://ncatlab.org/nlab/show/essential+ideal">current</a>

    Listed Rowen’s book, few other minor changes and new properties section. In particular, produced the following reasoning:

    A monomorphism $h$ is essential iff $g\circ h$ is monic only if $g$ is monic. Indeed, if $h:M\to N$ is essential and $g$ is not monic, then $Ker g\cap Im h\neq 0$ hence $g(Im h)\neq 0$ and $g\circ h$ is not monic. Conversely, suppose $g\circ h$ is monic implies $g$ is monic. If $h$ were not essential then there would be $0\neq K\subset N$ such that $K\cap M = 0$; in that case the cokernel map $g: N\to N/K$ is not monic while $g\circ h$ is monic because $Ker(g)\cap Im(h) = K\cap M = 0$ and $h$ is monic. This is a contradiction.

    diff, v6, current

11. • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeJul 5th 2024
    • PermaLink
    Author: zskoda
    Format: MarkdownItex[[socle|Socle]] of a module ([[internal sum]] of all [[simple module|simple]] submodules equals the intersection of all essential submodules. <a href="https://ncatlab.org/nlab/revision/diff/essential+ideal/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/essential+ideal/8">v8</a>, <a href="https://ncatlab.org/nlab/show/essential+ideal">current</a>

    Socle of a module (internal sum of all simple submodules equals the intersection of all essential submodules.

    diff, v8, current

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2024
    • PermaLink
    Author: Urs
    Format: MarkdownItexhave brushed-up wording and layout <a href="https://ncatlab.org/nlab/revision/diff/essential+ideal/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/essential+ideal/9">v9</a>, <a href="https://ncatlab.org/nlab/show/essential+ideal">current</a>

    have brushed-up wording and layout

    diff, v9, current