Reference request – ideals and normal subgroups in category theory

Question

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.

Answer 1

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.