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Thanks. To be linked to injective hull at some point.
Already, injective hull links to essential embedding, which redirects to essential ideal, but the meaning is different (although I'm sure related).
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?
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.
Oh, jeez. Yes; sorry. I can fix this a little later (but feel free to fix yourself now if you’d like).
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).
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.
Oh, no, I looked for it and missed it – we have a uniform module from before :) (I searched for uniform submodule that is why).
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.
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