I was wondering if, for example, we have a function $f(R)$ which maps all elements from the ring $R$ to another ring $S$, would the set which contains all elements mapped to $S$ be non-empty? This is because in the notes I took in one of my Algebra classes, where we had to prove that the set of the mapped elements is a also sub-ring, my professor wrote the following in order to initially prove that this set is not empty:
$f(R)\neq 0$ because $0_S \in f(R)$.