In my textbook they say that if $f: R \rightarrow S$ is a ring homomorphism, then:
if $I \subset R$ is an left ideal of $R$, then $f(I)$ is a left ideal of $S$.
However, I think that this is a mistake and that $f$ should be surjective for this to hold.
I have the following argument
Let $I$ be the identity map from $(\mathbb{Z},+,\cdot)$ to $(\mathbb{R},+,\cdot)$. It is known that $n\mathbb{Z}$ is an ideal from $(\mathbb{Z},+,\cdot)$. Thus, $I(n\mathbb{Z}) = n\mathbb{Z}$ should be an ideal of $(\mathbb{R},+, \cdot)$. However, since $\mathbb{R},+,\cdot$ is a field, it only has trivial left ideals, which contradicts that $n\mathbb{Z}$ is a left ideal, this because it is nontrivial.
So, my question is whether it is a mistake in my book or whether there is a hole in my argument?