Characteristic of a ring (2)
- Prove that for every commutative ring with unity,
, there is a unique ring homomorphism , and that for some unique nonnegative integer . The number is called the characteristic of and is denoted . - Suppose
and are fields for which there exists a ring homomorphism . Prove that .
View code
\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}