Question

(a) Show that there is exactly one maximal ideal in ${\mathrm{\mathbb{Z}}}_8$. Do the same for ${\mathrm{\mathbb{Z}}}_9$· [Hint: Exercise 6 in Section 6.1.]

Short answer

The maximal ideal in ${\mathrm{\mathbb{Z}}}_8$are $\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\mathrm{,}\left[\mathrm{6}\right]\right\}$ and the maximal ideal in${\mathrm{\mathbb{Z}}}_9$ are$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\mathrm{,}\left[\mathrm{6}\right]\right\}$.

Step-by-step solution

Determine set of ideal ℤ8

Firstly determine ideal set then find out the maximal ideal set.

Thus,

The set of ideal ${\mathrm{\mathbb{Z}}}_8$is given as,

$\left\{\left\{\left[\mathrm{0}\right]\right\}\mathrm{,}\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\mathrm{,}\left[\mathrm{6}\right]\right\}\mathrm{,}{\mathrm{\mathbb{Z}}}_{\mathrm{8}}\right\}$

Therefore, the only one maximal ideal is,

$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\mathrm{,}\left[\mathrm{6}\right]\right\}$.

Determine set of ideal ℤ9

Firstly determine ideal set then find out the maximal ideal set.

Thus,

The set of ideal ${\mathrm{\mathbb{Z}}}_9$is given as,

$\left\{\left\{\left[\mathrm{0}\right]\right\}\mathrm{,}\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\mathrm{,}\left[\mathrm{6}\right]\right\}\mathrm{,}{\mathrm{\mathbb{Z}}}_{\mathrm{9}}\right\}$

Therefore, the only one maximal ideal is,

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

Hence the maximal ideal in ${\mathrm{\mathbb{Z}}}_8$are $\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\mathrm{,}\left[\mathrm{6}\right]\right\}$and the maximal ideal in ${\mathrm{\mathbb{Z}}}_9$are $\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\mathrm{,}\left[\mathrm{6}\right]\right\}$.