Question

The following definition is needed for Exercises 41-43. Let R be a ring with identity. If there is a smallest positive integer n such that $\mathit{n}{\mathit{I}}_{\mathbf{R}}\mathbf{=}{\mathbf{0}}_{\mathbf{R}}$ , then R is said to have characteristic n. If no such nexists, R is said to have characteristic zero.

a) Show that role="math" localid="1647557577397" $\mathit{Z}$ has characteristic zero and positive integer n.

b) What is the characteristic of ${\mathit{Z}}_{\mathbf{4}}\mathbf{\times }{\mathit{Z}}_{\mathbf{6}}$.

Short answer

a) It is proved that $Z$ has characteristic 0 and a positive integer, $n$.

b)$\left[4,6\right]$ is the characteristic of $Z_4\times Z_6$.

Step-by-step solution

Show that Z has characteristic zero and positive integer n

a)

The set of positive integers is represented by $Z_{+}$ . In $Z$, for any $m\in Z_{+}$, there is $m1=m\ne 0$. As a result, $Z$possess characteristic 0.

There is $m\left[1\right]=m$ and $\left[m\right]=\left[0\right]$ in $Z_0$ such that if $n|m$ for every $m\in Z_{+}$.

Therefore, $Z_n$contains characteristics $n$because the smallest positive integer $m$ in which $n|m$ is $m=n$.

Thus, it is proved that $Z$ has characteristic 0 and a positive integer, $n$.

Determine the characteristic of Z4×Z6

In $Z_4\times Z_6$ for every $m\in Z_{+}$, there is $m\left(\left[1\right],\left[1\right]\right)=\left(\left[m\right],\left[m\right]\right)$ and $\left(\left[m\right],\left[m\right]\right)=\left(\left[0\right],\left[0\right]\right)$such that if $4|m$ and $6|m$is equal to $\left[4,6\right]|m$, namely $12|m$. Therefore, $Z_4\times Z_6$contains characteristic 12 because the smallest positive integer $m$ in which $12|m$ would be $m=12$.

Thus, $\left[4,6\right]$ is the characteristic of $Z_4\times Z_6$.