Question

(b) Show that${\mathrm{\mathbb{Z}}}_{10}$ and${\mathrm{\mathbb{Z}}}_{12}$ have more than one maximal ideal.

Short answer

The maximal ideals in${\mathrm{\mathbb{Z}}}_{10}$ are$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{5}\right]\right\}$ and$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{8}\right]\right\}$ and the maximal ideals in${\mathrm{\mathbb{Z}}}_{15}$ are$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{5}\right]\mathrm{,}\left[\mathrm{10}\right]\right\}$ and $\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{9}\right]\mathrm{,}\left[\mathrm{12}\right]\right\}$.

Step-by-step solution

Determine set of ideal ℤ10

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

Thus,

The set of ideal ${\mathrm{\mathbb{Z}}}_{10}$is given as,

$\left\{\left\{\left[\mathrm{0}\right]\right\}\mathrm{,}\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{5}\right]\right\}\mathrm{,}\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{8}\right]\right\}\mathrm{,}{\mathrm{\mathbb{Z}}}_{\mathrm{10}}\right\}$

Therefore, the maximal ideals are,

$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{5}\right]\right\}$and $\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{2}\right]\mathrm{,}\left[\mathrm{4}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{8}\right]\right\}$.

Determine set of ideal ℤ15

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

Thus,

The set of ideal ${\mathrm{\mathbb{Z}}}_{15}$is given as,

$\left\{\left\{\left[\mathrm{0}\right]\right\}\mathrm{,}\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{5}\right]\mathrm{,}\left[\mathrm{10}\right]\right\}\mathrm{,}\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{9}\right]\mathrm{,}\left[\mathrm{12}\right]\right\}\mathrm{,}{\mathrm{\mathbb{Z}}}_{\mathrm{15}}\right\}$

Therefore, the maximal ideals are,

$\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{5}\right]\mathrm{,}\left[\mathrm{10}\right]\right\}$and $\left\{\left[\mathrm{0}\right]\mathrm{,}\left[\mathrm{3}\right]\mathrm{,}\left[\mathrm{6}\right]\mathrm{,}\left[\mathrm{9}\right]\mathrm{,}\left[\mathrm{12}\right]\right\}$.