This is an exercise taken verbatim from Birkhoff and MacLane, A Survey of Modern Algebra:
Show that if $\phi: R \rightarrow R'$ is any homomorphism of rings, then the set $K$ of those elements in $R$ which are mapped onto $0 \in R'$ is a subring.
Let me recall from this section of the text the definition of a subring:
… define a subring of a commutative ring $A$ as a subset of $A$ which contains, with any two elements $f$ and $g$, also $f \pm g$ and $fg$, and which also contains the unity of $A$.
It seems to me that the implication in the exercise is false in general. I agree that $a,b \in K$ implies $a \pm b, ab \in K$, but it seems $1 \notin K$ in general. For example, take $\phi$ to be the identity map from $\mathbb{Z}$ to itself, then $K = \{0\}$ and hence $K$ does not contain the unity element. But the definition of subring given here requires that it contain the unity element.
I wonder if I am missing something here, and in case I am correct that the implication doesn't hold, is there some variation of the statement which is true?
Best Answer
This is a community wiki answer intended to remove this question from the unanswered queue.
As Daniel Fischer pointed out in the comments, the OP's reasoning is correct, and this was probably just a momentary lapse in consistency with definitions.