Characteristic of a ring (2)

#ring_theory

(a) Prove that for every commutative ring with unity, $R$ , there is a unique ring homomorphism $ϕ_{R} : Z \to R$ , and that $\ker (ϕ_{R}) = ⟨ d_{R} ⟩$ for some unique nonnegative integer $d_{R}$ . The number $d_{R}$ is called the characteristic of $R$ and is denoted $char (R)$ .
(b) Suppose $F_{1}$ and $F_{2}$ are fields for which there exists a ring homomorphism $f : F_{1} \to F_{2}$ . Prove that $char (F_{1}) = char (F_{2})$ .

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\begin{enumerate}[label=\alph*)]
	\item Prove that for every commutative ring with unity, $R$, there is a unique ring homomorphism $\phi_R: {\bf Z}\to R$, and that $\ker(\phi_R)=\langle d_R\rangle$ for some unique nonnegative integer $d_R$. The number $d_R$ is called the {\bfseries characteristic} of $R$ and is denoted $\operatorname{char}(R)$.
	\item Suppose $F_1$ and $F_2$ are fields for which there exists a ring homomorphism $f:F_1\to F_2$. Prove that $\operatorname{char}(F_1)=\operatorname{char}(F_2)$.
\end{enumerate}