**(a)** What does the question mean by 'determining the kernel'? I know that kernels are ideals and ideals are kernels. Since $\displaystyle \varphi$ is a ring homomorphism from $\displaystyle R \to \mathbb{Z}$, then $\displaystyle ker(\varphi)=\{ r \in R | \varphi (r) = 0 \}$ is an ideal of R.

Now, a subring I is an ideal of R if $\displaystyle rI := \{ ra: a \in I \} \subseteq I$ and $\displaystyle Ir := \{ ar : a \in I \} \subseteq I$ for $\displaystyle \forall r \in R$. (The subring I is closed under multiplication with arbitrary elements of R). Is this all I need to say?!