Question

List the distinct principal ideals in $Z_2\times Z_3$

Short answer

Principal ideals in $Z_2\times Z_3$ are

$\begin{array}{rcl}\left(\left(0,0\right)\right)& =& \left\{\left(0,0\right)\right\}\\ \left(\left(1,0\right)\right)& =& \left\{\left(1,0\right),\left(0,0\right)\right\}\\ \left(\left(0,1\right)\right)& =& \left(\left(0,2\right)\right)=\left\{\left(0,0\right),\left(0,1\right)\left(0,2\right)\right\}\\ \left(\left(1,1\right)\right)& =& \left(\left(1,2\right)\right)={\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_3\end{array}$

Step-by-step solution

Definition

Let $R$be a commutative ring with identity,$c\in R$ , and$I$ the set of all multiples of$c$ in$R$ : that is,$I=\left\{rc\overline{)r\in R}\right\}$ Then$I$ is an ideal.

Here, the ideal $I$is called the principal ideal generated by$c$ and is denoted by$\left(c\right)$ .

Find ideals generated by each element of  Z5

${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_3=\left\{\left(0,0\right),\left(1,0\right),\left(0,1\right),\left(0,2\right),\left(1,1\right),\left(1,2\right)\right\}$

$\begin{array}{rcl}\left(\left(0,0\right)\right)& =& \left\{\left(0,0\right)\right\}\\ \left(\left(1,0\right)\right)& =& \left\{\left(1,0\right),\left(0,0\right)\right\}\\ \left(\left(0,1\right)\right)& =& \left\{\left(0,0\right),\left(0,1\right),\left(0,2\right)\right\}\\ \left(\left(0,2\right)\right)& =& \left\{\left(0,0\right),\left(0,1\right),\left(0,2\right)\right\}\\ \left(\left(1,1\right)\right)& =& \left\{\left(0,0\right),\left(1,0\right),\left(0,1\right),\left(0,2\right),\left(1,1\right),\left(1,2\right)\right\}\\ \left(\left(1,2\right)\right)& =& \left\{\left(0,0\right),\left(1,0\right),\left(0,1\right),\left(0,2\right),\left(1,1\right),\left(1,2\right)\right\}\\ \left(\left(0,0\right)\right)& =& \left\{\left(0,0\right)\right\}\\ \left(\left(1,0\right)\right)& =& \left\{\left(1,0\right),\left(0,0\right)\right\}\\ \left(\left(0,1\right)\right)& =& \left\{\left(0,2\right),\left(0,0\right),\left(0,1\right),\left(0,2\right)\right\}\\ \left(\left(1,1\right)\right)& =& \left(\left(1,2\right)\right)={\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_3\end{array}$

.

Conclusion

Distinct principal ideals in${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_3$ are

$$\begin{gathered} \left(\left(0,0\right)\right)=\left\{\left(0,0\right)\right\} \\ \left(\left(1,0\right)\right)=\left\{\left(1,0\right),\left(0,0\right)\right\} \\ \left(\left(0,1\right)\right)=\left(\left(0,2\right)\right)\left\{\left(0,0\right),\left(0,1\right),\left(0,2\right)\right\} \\ \left(\left(1,1\right)\right)=\left(\left(1,2\right)\right)={\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_3 \end{gathered}$$