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**Formal Definition**

An **isomorphism** $\phi :R\rightarrow R'$ from a ring $R$ to a ring $R'$ is a homomorphism that is one to one and onto $R'$. The rings $R$ and $R'$ are then **isomorphic**.

**Informal Definition**

Replace this text with an informal definition.

**Example(s)**

Let $gcd(r,s)=1$, then the rings $Z_r \times Z_s$ and $Z_{rs}$ are isomorphic.

**Non-example(s)**

As abelian groups, $\langle \mathbb{Z}, +\rangle$ and $\langle \mathbb{2Z}, +\rangle$ are isomorphic under the map $\phi :\mathbb{Z}\rightarrow\mathbb{Z}$, with $\phi (x) =2x$ for $x\in \mathbb{Z}$. Here $\phi$ is not a ring isomorphism, for $\phi(xy)=2xy$,while $\phi (x)\phi(y)=2x2y=4xy$.

**Additional Comments**

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