Question

a) Show that the set of non-units in${\mathrm{\mathbb{Z}}}_8$ is an ideal.

b) Do part (a) for${\mathrm{\mathbb{Z}}}_9$ [Also, see Exercise 24.]

Short answer

It is concluded the set of non-units in ${\mathrm{\mathbb{Z}}}_8$is ideal.

Step-by-step solution

Determine that the set of non-units in ℤ8 is ideal

Using theorem 2.10, this is denoted by $N$.

$N=\left\{\left[0\right],\left[2\right],\left[4\right],\left[6\right]\right\}$

From the above equation,

$N=\left\{m\left[2\right]\overline{)m\in {\mathrm{\mathbb{Z}}}_8}\right\}$

This can be checked by direct computation.

Therefore, using theorem 6.2, notice that ${\mathrm{\mathbb{Z}}}_8$is truly commutative ring with identity, and the identity is $\left[1\right]$.

Therefore, it is concluded that $N$is ideal.

Determine that the set of non-units in ℤ9 is an ideal

Using theorem 2.10, this is denoted by $N$.

$N_2=\left\{\left[0\right],\left[3\right],\left[6\right]\right\}$

From the above equation,

$N_2=\left\{m\left[3\right]\overline{)m\in {\mathrm{\mathbb{Z}}}_9}\right\}$

This can be checked by direct computation.

Therefore, using Theorem 6.2, ${\mathrm{\mathbb{Z}}}_9$is truly a commutative ring with identity, and the identity is $\left[1\right]$.

Therefore, it is concluded that $N$is ideal.