Question

List all maximal ideals in ${\mathrm{\mathbb{Z}}}_6$. Do the same in ${\mathrm{\mathbb{Z}}}_{12}$.

Short answer

The maximal ideals in ${\mathrm{\mathbb{Z}}}_6$are $\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\right\}$and $\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\right\}$.

The maximal ideals are in ${\mathrm{\mathbb{Z}}}_{12}$are$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{9}\right]\right\}$. and$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{8}\right]\mathrm{,}\left[\mathrm{10}\right]\right\}$.

Step-by-step solution

Determine set of ideal ℤ6

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

Thus,

The set of ideal ${\mathrm{\mathbb{Z}}}_6$is given as,

$\left\{\left\{\left[\mathrm{0}\right]\right\}\mathrm{,}\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\right\}\mathrm{,}\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\right\}\mathrm{,}{\mathrm{\mathbb{Z}}}_{\mathrm{6}}\right\}$

Therefore, the maximal ideals are,

$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\right\}$and $\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\right\}$.

Determine set of ideal ℤ12

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

Thus,

The set of ideal ${\mathrm{\mathbb{Z}}}_{12}$is given as,

$\left\{\left\{\left[\mathrm{0}\right]\right\}\mathrm{,}\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{6}\right]\right\}\mathrm{,}\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{4}\right]\mathrm{,}\left[\mathrm{8}\right]\right\}\mathrm{,}\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{9}\right]\right\}\mathrm{,}\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{8}\right]\mathrm{,}\left[\mathrm{10}\right]\right\}\mathrm{,}{\mathrm{\mathbb{Z}}}_{\mathrm{12}}\right\}$

Therefore, the maximal ideals are,

$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{9}\right]\right\}$and $\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{8}\right]\mathrm{,}\left[\mathrm{10}\right]\right\}$.

Hence the maximal ideals in${\mathrm{\mathbb{Z}}}_6$ are$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\right\}$ and .$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\right\}$The maximal ideals are in${\mathrm{\mathbb{Z}}}_{12}$ are$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{9}\right]\right\}$ and $\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{8}\right]\mathrm{,}\left[\mathrm{10}\right]\right\}$.