Question

Let$\mathit{I}$ be the set of elements of$\mathit{T}$ whose numerators are divisible by $\mathit{p}$.

Prove that$\mathit{I}$ is an ideal in $\mathit{T}$.

Short answer

It is proved that $I$is an ideal by theorem 6.2.

Step-by-step solution

Obtain that I is an ideal

Remember that $\left(p\right)\underline{\subset }I$.

If$\frac{a}{b}\in T$ for $K\in \square$.

It has$a=pk$ and such that$\frac{a}{b}=p\frac{k}{b}\in \left(p\right)$

Hence,$I=\left(p\right)$

Use theorem 6.2

And by using theorem 6.2, $I$is an ideal.