Unique homomorphic extension theorem

The Unique Homomorphic Extension Theorem is a result in the field of algebra, particularly concerning rings and homomorphisms. It typically states that if you have a ring \( R \) and a subring \( S \), along with a homomorphism defined on \( S \), then there exists a unique (in the case of certain conditions) homomorphic extension of this mapping up to the whole ring \( R \).