Answer to Question #104395 in Abstract Algebra for Venkatesh

Question

Z ring of integers, R ring, f: ring homomorphism from Z to R which is injective. Also if n is not in 13Z, then f(n) is unit in R. Can R be integral domain. One example of such a map.

Answer 1

The ring $R$ can be an integral domain. Let $R$ be $\mathbb{Q}$ and $f$ be defined by $f(n) = n$. Obviously, $f$ is injective and a ring homomorphism.

Assume that $n\not\in 13 \mathbb{Z}$. Since $0\in 13 \mathbb{Z}$, $n\neq 0$. Then $f(n)\neq 0$, so $f(n)$ is a unit because $\mathbb{Q}$ is a field. Every field is an integral domain, hence $R$ is an integral domain.