Isomorphism Of Ring

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 $\phi (x)\phi(y)=2x2y=4xy$.