Definition of an Isomorphism

An isomorphism from $R$ to $S$ is a mapping $\theta :R\to S$ that is…

• Injective: $\theta (x)=\theta (y)⟹x=y$.
• Surjective: $\forall s\in S,\exists r\in R:\theta (r)=s$.
• Homomorphic: $\theta (x+y)=\theta (x)+\theta (y),\theta (xy)=\theta (x)\theta (y)$.

If such isomorphism exists, we write $R\cong S$; $R$ and $S$ are isomorphic.