Question

Let$\mathit{I}$ be the set of elements of $\mathit{S}$ with even numerators. Prove that $\mathit{I}$ is an ideal in $\mathit{S}$.

Short answer

It is proved$I$is an ideal in$S$ by Theorem 6.2.

Step-by-step solution

Write the condition

Recall that $\left(2\right)\subseteq I$

Also, $\frac{a}{b}\in I\subseteq S$for $k\in \mathbb{Z}$

     Step 2: Show that isI is an ideal in S

If $a=2k$, such that $\frac{a}{b}=2\frac{k}{b}\in \left(2\right)$

Thus, $I=\left(2\right)$.

Using Theorem 6.2, $I$is an ideal in localid="1649245208971" $S$.

Hence, it is proved$I$is an ideal in$S$by Theorem 6.2.