Solution
= Solution
The <first isomorphism theorem for rings> states that the map
$$
\boxed{R/\ker f\longrightarrow\operatorname{im}f,\qquad r+\ker f\longmapsto f(r)}
$$
is a <ring isomorphism>. It is well defined because two representatives differ by an element of the <kernel> exactly when their images agree. It preserves addition and multiplication, is onto its stated codomain, and has trivial <kernel>. In particular, if $f$ is <surjective> then $R/\ker f\cong S$.