Ring isomorphism - Knowlpedia

A ring isomorphism is a bijective ring homomorphism $\varphi:R\to S$ . Its inverse function $\varphi^{-1}:S\to R$ is then automatically a ring homomorphism.

Remarks

Isomorphic rings have corresponding ideal lattices, unit groups, and ring-theoretic invariants.

Examples

The map $R[x]/(x)\to R$ sending $f(x)+(x)\mapsto f(0)$ is a ring isomorphism.
For a commutative ring $R$ , $R\times R \cong R[t]/(t(t-1))$ .
The inclusion $\mathbb Z\hookrightarrow\mathbb Q$ is a ring homomorphism but not an isomorphism because it is not surjective.