Question

List the distinct principal ideals in each ring :

1. $Z_5$
   
2. $Z_9$
3. $Z_{12}$
   

Short answer

(a) Principal ideals in ${\mathrm{\mathbb{Z}}}_5$are

$$\begin{gathered} \left(0\right) \\ \left(1\right)=\left(2\right)=\left(3\right)=\left(4\right)={\mathrm{\mathbb{Z}}}_5 \end{gathered}$$

(b) Principal ideals in ${\mathrm{\mathbb{Z}}}_9$are

$$\begin{gathered} \left(0\right) \\ \left(1\right)=\left(2\right)=\left(4\right)=\left(5\right)=\left(7\right)=\left(8\right)={\mathrm{\mathbb{Z}}}_9 \\ \left(3\right)=\left(6\right)=\left\{0,3,6\right\} \end{gathered}$$

(c) Principal ideals in ${\mathrm{\mathbb{Z}}}_{12}$are

$$\begin{gathered} \left(0\right)=\left\{0\right\} \\ \left(1\right)=\left(5\right)=\left(7\right)=\left(11\right)={\mathrm{\mathbb{Z}}}_{12} \\ \left(2\right)=\left(10\right)=\left\{0,2,4,6,8,10\right\} \\ \left(3\right)=\left(9\right)=\left\{0,3,6,9\right\} \\ \left(4\right)=\left(8\right)=\left\{0,4,8\right\} \\ \left(6\right)=\left\{0,6\right\} \end{gathered}$$

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 role="math" localid="1648701448491" $I=\left\{rc\overline{)r\in R}\right\}$ Then $I$is an ideal.

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

The ideals generated by each element of $Z_5$are

$$\begin{gathered} Z_5=\left\{0,1,2,3,4\right\} \\ \left(0\right)=\left\{0\right\} \\ \left(1\right)=\left\{0,1,2,3,4\right\} \\ \left(2\right)=\left\{0,2,4,1,3\right\} \\ \left(3\right)=\left\{0,3,1,4,2,\right\} \\ \left(4\right)=\left\{0,4,3,2,1\right\} \\ \left(1\right)=\left(2\right)=\left(3\right)=\left(4\right)={\mathrm{\mathbb{Z}}}_5 \end{gathered}$$

The distinct principal ideals of${\mathrm{\mathbb{Z}}}_5$ are

$\left(0\right)=\left\{0\right\}and\left(1\right)=\left(2\right)=\left(3\right)=\left(4\right)={\mathrm{\mathbb{Z}}}_5$

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 $\left(c\right)$.

The ideals generated by each element of$Z_9$

$$\begin{gathered} {\mathrm{\mathbb{Z}}}_9=\left\{0,1,2,3,4,5,6,7,8,\right\} \\ \left(0\right)=\left\{0\right\} \\ \left(1\right)=\left\{0,1,2,3,4,5,6,7,8\right\} \\ \left(2\right)=\left\{0,2,4,6,8,1,3,5,7\right\} \\ \left(3\right)=\left\{0,3,6\right\} \\ \left(4\right)=\left\{0,4,8,3,7,2,6,1,5\right\} \\ \left(5\right)=\left\{0,5,1,6,2,7,6,8,4\right\} \\ \left(6\right)=\left\{0,6,3\right\} \\ \left(7\right)=\left\{0,7,5,3,1,8,6,4,2\right\} \\ \left(8\right)=\left\{0,8,7,6,5,4,3,2,1\right\} \\ \left(1\right)=\left(2\right)=\left(4\right)=\left(5\right)=\left(7\right)=\left(8\right)={\mathrm{\mathbb{Z}}}_9 \\ \left(3\right)=\left(6\right)=\left\{0,3,6\right\} \end{gathered}$$

Distinct principal ideals of ${\mathrm{\mathbb{Z}}}_9$are

$$\begin{gathered} \left(0\right) \\ \left(1\right)=\left(2\right)=\left(4\right)=\left(5\right)=\left(7\right)=\left(8\right)={\mathrm{\mathbb{Z}}}_9 \\ \left(3\right)=\left(6\right)=\left\{0,3,6\right\} \end{gathered}$$

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 $\left(c\right)$.

The ideals generated by each element of ${\mathrm{\mathbb{Z}}}_{12}$

$$\begin{gathered} {\mathrm{\mathbb{Z}}}_{12}=\left\{0,1,2,3,4,5,6,7,8,9,10,11,\right\} \\ \left(0\right)=\left\{0\right\} \\ \left(1\right)=\left\{0,1,2,3,4,5,6,7,8,9,10,11,\right\} \\ \left(2\right)=\left\{0,2,4,6,8,10\right\} \\ \left(3\right)=\left\{0,3,6,9\right\} \\ \left(4\right)=\left\{0,4,8\right\} \\ \left(5\right)=\left\{0,5,10,3,8,1,6,11,4,9,2,7\right\} \\ \left(6\right)=\left\{0,6\right\} \\ \left(7\right)=\left\{0,7,2,9,4,11,6,1,8,3,10,5\right\} \\ \left(8\right)=\left\{0,8,4\right\} \\ \left(9\right)=\left\{0,9,6,3\right\} \\ \left(10\right)=\left\{0,10,8,6,4,2\right\} \\ \left(11\right)=\left\{0,11,10,9,8,7,6,5,4,3,2,1\right\} \end{gathered}$$

Thus,

$$\begin{gathered} \left(1\right)=\left(5\right)=\left(7\right)=\left(11\right)={\mathrm{\mathbb{Z}}}_{12} \\ \left(2\right)=\left(10\right)=\left\{0,2,4,6,8,10\right\} \\ \left(3\right)=\left(9\right)=\left\{0,3,6,9\right\} \\ \left(4\right)=\left(8\right)=\left\{0,4,8\right\} \\ \left(6\right)=\left\{0,6\right\} \end{gathered}$$

Distinct principal ideals of ${\mathrm{\mathbb{Z}}}_{12}$are

$$\begin{gathered} \left(0\right)=\left\{0\right\} \\ \left(1\right)=\left(5\right)=\left(7\right)=\left(11\right)={\mathrm{\mathbb{Z}}}_{12} \\ \left(2\right)=\left(10\right)=\left\{0,2,4,6,8,10\right\} \\ \left(3\right)=\left(9\right)=\left\{0,3,6,9\right\} \\ \left(4\right)=\left(8\right)=\left\{0,4,8\right\} \\ \left(6\right)=\left\{0,6\right\} \end{gathered}$$