# Reference request - ideals and normal subgroups in category theory

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.

• 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). – Berci Feb 26 at 20:58
• You may find of interest my post on ideal-determined varieties. – Bill Dubuque Feb 26 at 23:26