I am reading the book Introduction to commutative algebra by Atiyah and Macdonald. On page 50, Line -7, it is said that "if $f: A \to B$ and $\mathfrak{q}$ is a primary ideal in $B$, then $A/\mathfrak{q}^c$ is isomorphic to a subring of $B/\mathfrak{q}$". How to prove this result? Thank you very much.
$\def\p{\mathfrak p}\def\q{\mathfrak q}$Let $\pi \colon B \to B/\q$ denote the projection, consider $\pi f \colon A \to B/\q$. $\pi f$ is a homomorphism with kernel $f^{-1}(\q) = \q^c$. So we get a monomorphism $g \colon A/\q^c \to B/\q$ by $g(a + \q^c) = f(a) + \q$. As $g$ is one-to-one, we may regard $A/\q^c$ as a subring of $B/\q$.
