Ring isomorphism

A ring isomorphism is a ring homomorphism $\varphi:R\to S$ that is bijective and whose inverse function $\varphi^{-1}:S\to R$ is also a ring homomorphism.

Equivalently, $\varphi$ is an isomorphism iff it is bijective and respects the ring operations; then $\varphi^{-1}$ exists as an inverse function and automatically preserves operations. Isomorphic rings are “the same” for algebraic purposes: they have corresponding ideal lattices, unit groups, and invariants.

Examples:

• The map $R[x]/(x)\to R$ sending $f(x)+(x)\mapsto f(0)$ is a ring isomorphism.
• For a commutative ring $R$, $R\times R \cong R[t]/(t(t-1))$ (one concrete model of a product ring).
• The map $\mathbb Z\to \mathbb Z$, $n\mapsto 2n$, is a ring homomorphism but not an isomorphism (not surjective).