Characteristic of a ring

#ring_theory

Let $R$ be a commutative ring with $1$ . The characteristic $char (R)$ of $R$ is the unique integer $n \geq 0$ such that $⟨ n ⟩ \subset Z$ is the kernel of the homomorphism $θ : Z \to R$ given by

θ (m) = {\begin{cases} \underset{m}{\underset{⏟}{1_{R} + \dots + 1_{R}}}, & if m \geq 0 \\ \underset{| m |}{\underset{⏟}{- 1_{R} + \dots + - 1_{R}}}, & if m < 0 \end{cases}

(a) Prove that if $f : R \to S$ is a monomorphism of commutative rings with $1$ , then $char (R) = char (S)$ .
(b) Prove by given an example that $char (R)$ is not always preserved by ring homomorphisms.

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Let $R$ be a commutative ring with $1$. The {\bfseries characteristic} $\operatorname{char}(R)$ of $R$ is the unique integer $n\geq 0$ such that $\langle n\rangle \subset {\bf Z}$ is the kernel of the homomorphism $\theta:{\bf Z}\to R$ given by
\[
	\theta(m)=\begin{cases} \underbrace{1_R+\cdots +1_R}_{m}, & \text{ if }m\geq 0 \\ \underbrace{-1_R+\cdots+-1_R}_{|m|}, & \text{ if }m<0\end{cases}
\]
\begin{enumerate}[label=\alph*)]
	\item Prove that if $f:R\to S$ is a monomorphism of commutative rings with $1$, then $\operatorname{char}(R)=\operatorname{char}(S)$.
	\item Prove by given an example that $\operatorname{char}(R)$ is not always preserved by ring homomorphisms.
\end{enumerate}