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.