In Lindsay Childs' Algebra text (3ed pg 141), it statements this proposition regarding a ring homomorphism: $ Let f: R \rightarrow S$ be a homomorphism where R is a field and $1 \ne 0$ in $S$. Then $f$ is one-to-one.
My confusion is about $1 \ne 0$. Is that possible? I can only see that in congruence where $[1]_{1} = [0]_{1}$, but again, congruence class of integers mod 1 is not really interesting... btw, the book's errata says nothing about this page.
Any help would be appreciated! Thanks!