First Isomorphism Theorem

Preface: This theorem applies for Groups, Rings, Modules, Algebras, and any other algebraic structure where you see the word "homomorphism". It is a categorical result i.e. it is independent of the structure used. Rings will be used in the theorem statement.

The First Isomorphism Theorem: Let $$\alpha: R\rightarrow S$$ be a ring homomorphism. Then

$$ R/ker(\alpha) \cong im(\alpha)$$