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I'm looking for a categorial notion of a normal subgroup and an ideal (of a ring, and a non-associative algebra).

Basing on the following observations:

  • they are used to define quotient objects in the category of groups and commutative rings,
  • given a connected Lie group, there is a correspondence between its normal connected Lie subgroups and ideals of its Lie algebra,

I expect that there may exist a construction in category theory generalizing these notions.

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  • $\begingroup$ Yes, $A\hookrightarrow B$ is (isomorphic to) an embedding of an ideal of $B$ iff it is a kernel of some $g:B\to C$ (that is, the equalizer of $g$ and the trivial morphism). $\endgroup$ – Berci Feb 26 at 20:58
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    $\begingroup$ You may find of interest my post on ideal-determined varieties. $\endgroup$ – Bill Dubuque Feb 26 at 23:26
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What you want is the notion of a congruence. It's a convenient fact about groups resp. rings that congruences are equivalent to normal subgroups resp. (two-sided) ideals; in general, for example when dealing with monoids, you really need to work with congruences.

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