Identity of the kernel of a ring homomorphism

Question

Say we have a ring homomorphism from a ring $R$ to $S$ with multiplicative identities $1_R$ and $1_S$ respectively, is it true that the identity of the kernel which is a subring of $R$ is also $1_R$ if and only if $1_R$ maps to the $0$ of ring $S$ (the additive identity of $S$). Or otherwise the kernel doesn’t have an identity. Is that correct? Thanks in advance

Answer 1

Take $R=S=\mathbb Z_6$ and the ring homomorphism $f$ given by $x \mapsto 4x$.

Then $\ker f = \{0,3\}$ is a ring with $3$ as an identity.

($f$ does not send $1$ to $1$ but it is multiplicative.)

Answer 2

Let $J$ be the kernel of $f\colon R\to S$. If $1_R\in J$, then clearly $f=0$.

In theory, $J$ might have an identity $1_J$ which is not an identity for all of $R$.