Exercise 3.12

The ‘characteristic’ of a ring $R$ is the smallest positive integer $n$ such that $n(1_R)=0_R$, or $0$ if there is no such positive integer.

• Show that the characteristic of $\mathbb{Z}$ is 0, and the characteristic of $\mathbb{Z}/m\mathbb{Z}$ is $m$.
• Show that if $R$ has characteristic $n$, then $na=0_R$ for all elements $a$ of $R$.

Answer

In $\mathbb{Z}$, if $n>0$, then $n\cdot 1=n>0$. Thus, there is no such positive integer such that $n\cdot 1=0$. Therefore, the characteristic is $0$. In $\mathbb{Z}/m\mathbb{Z}$, $m[1]=[m]=[0]$. Suppose there is another positive integer $n<m$ for which $n[1]=[0]$. Then, $[n]=[0]$, so $m$ must divide $n$, but $0<n<m$ so this is not possible. Thus, the characteristic is $m$.

Let $R$ be a ring with characteristic $n$. Then, by Exercise 3.8, we have $na=(n{1}_R)\cdot a$. Since $n$ is the characteristic of $R$. $n(1_R)=0_R$. Thus, $na=0_{R}a=0_R$. $■$