From the 2nd edition, page 189.
If $f : A \to B$ is a homomorphism from $A$ onto $B$, and $B$ is a field, then the kernel of $f$ is a maximal ideal.
Let $K$ be the kernel. Pinter defines a maximal ideal as proper (page 189), i.e., not $A$ or $\{0\}$. He also allows $\{0\}$ to be a field (page 172). Thus, if $B = \{0\}$, then $K = A$, and so $K$ is not a maximal ideal.
From the 2nd edition, page 189.
If$f : A \to B$ is a homomorphism from $A$ onto $B$ , and $B$ is a field, then the kernel of $f$ is a maximal ideal.
Let$K$ be the kernel. Pinter defines a maximal ideal as proper (page 189), i.e., not $A$ or $\{0\}$ . He also allows $\{0\}$ to be a field (page 172). Thus, if $B = \{0\}$ , then $K = A$ , and so $K$ is not a maximal ideal.