Question

Show that there are infinitely many integral domains R such that $\mathrm{\mathbb{Z}}⊑R⊑\mathrm{\mathbb{Q}}$, each of which has $\mathrm{\mathbb{Q}}$ as its field of Quotient. [Hint: Exercise 28 in Section 3.1.]

Short answer

It is proved that there are infinitely many integral domains R such that$\mathrm{\mathbb{Z}}⊑R⊑\mathrm{\mathbb{Q}}$ .

Step-by-step solution

Use Exercise 3.1.28

For every prime number p, we have$R_p=\left\{\frac{r}{p^i}\in \mathrm{\mathbb{Q}}/r\in \mathrm{\mathbb{Z}},i\in {\mathrm{\mathbb{Z}}}_{>0}\right\}$ is a subring of$\mathrm{\mathbb{Q}}$ that contains$\mathrm{\mathbb{Z}}$ as a subring.

If we prove that for$p\ne q$ distinct primes, we have $R_p\ne R_q$. Then, we will have infinitely many distinct integral domains between$\mathrm{\mathbb{Z}}$ and$\mathrm{\mathbb{Q}}$ .

Prove that Rp≠Rq

Suppose there are$r\in \mathrm{\mathbb{Z}}$ and$i\in {\mathrm{\mathbb{Z}}}_{>0}$ such that$\frac{1}{p}=\frac{r}{q^i}$ .

Then, $q^i=rp$

This implies that, $p/q^i$ and p is prime that $p/q$, which is a contradiction.

Therefore, $\frac{1}{p}\in R_q$and so $R_p\ne R_q$.

Therefore, there are infinitely many distinct integral domains between $\mathrm{\mathbb{Z}}$and $\mathrm{\mathbb{Q}}$.

Hence proved.