Question

Question: What is the characteristic of

$$\begin{gathered} \mathbf{\text{(a)}}\mathbf{\mathbb{Q}} \\ \mathbf{\text{(b)}}{\mathbf{\mathbb{Z}}}_{\mathbf{2}}\mathbf{\times }{\mathbf{\mathbb{Z}}}_{\mathbf{6}} \\ \mathbf{\text{(c)}}{\mathbf{\mathbb{Z}}}_{\mathbf{3}}\left[x\right] \\ \mathbf{\text{(d)}}\mathit{M}\left(ℝ\right) \\ \mathbf{\text{(e)}}\mathit{M}\left({\mathbb{Z}}_3\right) \end{gathered}$$

Short answer

Answer:

1. The characteristic of$\mathrm{\mathbb{Q}}$is zero.
2. The characteristic of$\left({\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_6\right)$ is 6.
3. The characteristic of ${\mathrm{\mathbb{Z}}}_3\left[x\right]$is 3.
4. The characteristic of $M\left[\mathrm{ℝ}\right]$is zero.
5. The characteristic of $M\left[{\mathrm{\mathbb{Z}}}_3\right]$is 3.

Step-by-step solution

Describe the ring  

A ring is fundamental algebraic structure. It consists of a set equipped with two binary operations that generalized the arithmetic operations of addition and multiplications.

Find the characteristic of ℚ(a) 

Since there does not exist any number positive integer n such that na = 0 for all$a\in \mathrm{\mathbb{Q}}$.

As if there exists a positive integer such that n such that for all in R is said to be of finite characteristic.

The least positive integers is called the characteristic of R and n = 0 such number does not exist therefore, na = 0 only if n = 0 .

Therefore, the characteristic of $\mathrm{\mathbb{Q}}$is zero.

Find the characteristic of ℤ2×ℤ6(b)

Since$\text{char}\left({\mathrm{\mathbb{Z}}}_n\times {\mathrm{\mathbb{Z}}}_m\right)=\text{LCM}\left(n,m\right)$.

For this case,

role="math" $$\begin{gathered} \text{char}\left({\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_6\right)=\text{LCM}\left(2,6\right) \\ =6 \end{gathered}$$

Therefore, the characteristic of $\left({\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_6\right)$is 6.

Find the characteristic of ℤ3[x](c) 

Since .

$\text{char}\left({\mathrm{\mathbb{Z}}}_n\left[x\right]\right)=n$

For this case, n = 3 so,

$\text{char}\left({\mathrm{\mathbb{Z}}}_n\left[x\right]\right)=3$

Therefore, the characteristic$\left({\mathbb{Z}}_n\left[x\right]\right)$ of is 3 .

 Step 5: Find the characteristic of Mℝ(d)

As if there exists a positive integer such that for all in is said to be of finite characteristic.

The least positive integer is called the characteristic of M(R), and such a number does not exist therefore, na = 0 only if n = 0 since $M\left[\mathrm{ℝ}\right]$is infinite.

Therefore, the characteristic of M(R) is zero.

Find the characteristic of Mℤ3(e) 

As if there exists a positive integer such that n such that na = 0 for all R in is said to be of finite characteristic. The least positive integer is called the characteristic of R, and such number does not exist, therefore, only if since $M\left[{\mathrm{\mathbb{Z}}}_3\right]$is finite.

Therefore, the characteristic of $M\left[{\mathrm{\mathbb{Z}}}_3\right]$is 3.