Question

Show that the subset$S=\left\{0,2,4,6,8\right\}$ of${\mathrm{\mathbb{Z}}}_{10}$ is a subring. Does$S$ have an identity?

Short answer

The addition and multiplication of the subset$S$ shows that it satisfies the condition of Theorem 3.2. Therefore, the subset$S$ of${\mathrm{\mathbb{Z}}}_{10}$ is a subring.

Step-by-step solution

Addition of Subset

Conditions of Theorem 3.2

• The subset $S$ is closed under addition.
• The subset$S$ is closed under multiplication.
• $O_{{\mathrm{\mathbb{Z}}}_{10}}\in S$
• If $a\in S$, then the solution of the equation $a+x=O_{{\mathrm{\mathbb{Z}}}_{10}}$ is in$S$ .

Then the subset $S$ is a subring of ${\mathrm{\mathbb{Z}}}_{10}$.

Consider the given subset:

role="math" localid="1646374851107" $S=\left\{0,2,4,6,8\right\}\subseteq {\mathrm{\mathbb{Z}}}_{10}$

Taking addition of Subset

+	0	2	4	6	8
0	0	2	4	6	8
2	2	4	6	8	0
4	4	6	8	0	2
6	6	8	0	2	4
8	8	0	2	4	6

The subset$S$ is closed under addition.

Multiplication of Subset

$S=\left\{0,2,4,6,8\right\}\subseteq {\mathrm{\mathbb{Z}}}_{10}$

Taking multiplication of Subset $S$

+	0	2	4	6	8
0	0	0	0	0	0
2	0	4	8	2	6
4	0	8	6	4	2
6	0	2	4	6	8
8	0	6	2	8	4

The subset $S$ is closed under multiplication.

Result

Here, $S$ is subring of${\mathrm{\mathbb{Z}}}_{10}$ .

As shown above, the subset$S$ satisfies all the conditions of Theorem 3.2, that is,

1. The subset$S$ is closed under addition.
2. The subset $S$ is closed under multiplication.
3. $O_{{\mathrm{\mathbb{Z}}}_{10}}\in S$
4. If$a\in S$ , then the solution of the equation$a+x=O_{{\mathrm{\mathbb{Z}}}_{10}}$ is in S .

As 0+0=0, 2+8=0, 6+4=0 thus$S$ is cumulative.

Therefore,$S$ is a subring of ${\mathrm{\mathbb{Z}}}_{10}$.