# Kernel of a congruence

From Groupprops

## Definition

Let be a variety of algebras with zero and be an algebra in . Then, a nonempty subset of is termed the **kernel of a congruence** if it satisfies the following equivalent conditions:

- There exists a congruence on such that is the congruence class of the zero element.
- There exists a surjective homomorphism of algebras such that is the inverse image under of the zero element of .
- There exists a homomorphism (not necessarily surjective) of algebras such that is the inverse image under of the zero element of .