(This question originates from Pinter's Abstract Algebra Chapter 18 exercise I2.)
Let $A$ be a ring. Prove that if $f: A\rightarrow B$ is a homomorphism from $A$ onto $B$, and $B$ is a field, then the kernel of $f$ is a maximal ideal.
Proof attempt:
Let $K$ be the kernel of $f$. By the fundamental homomorphism theorem, $A/K\cong B$, so $B$ being a field implies $A/K$ is a field. A field $F$ can have no ideals except $\{0\}$ and $F$. Therefore the kernel of $f$, $K$ in $A/K$, is a maximal ideal.
Does this look right?