query: how can i prove that every subring of $\mathbb{Q}$ is PID?
positive_passages.docid: 21747
positive_passages.text: A subring of the field of fractions of a PID is a PID as well.
negative_passages.docid: 136855
negative_passages.text: Proof that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ is a PID
negative_passages.docid: 24561
negative_passages.text: $\mathbb S_n$ as semidirect product
negative_passages.docid: 152297
negative_passages.text: Understanding quotients of $\mathbb{Q}[x]$
negative_passages.docid: 157420
negative_passages.text: Question about Subrings and integraly closed
negative_passages.docid: 88759
negative_passages.text: Proofs of Sylow theorems
negative_passages.docid: 119360
negative_passages.text: Find all subrings of $\mathbb{Z}^2$
negative_passages.docid: 55026
negative_passages.text: $\mathbb{Q}[X,Y]/(Y^2-X^3)$ is not a UFD
negative_passages.docid: 134840
negative_passages.text: Generators and relations for $\text{PSL}(2, \mathbb{F}_q)$
negative_passages.docid: 136770
negative_passages.text: Any field that contains the integers contains the rationals as a subfield (Analysis proof)
negative_passages.docid: 157496
negative_passages.text: Determine the Galois group of $\mathbb{Q}(\sqrt{a+b\sqrt{d}})$
negative_passages.docid: 27629
negative_passages.text: How to prove that $S^{2n+1}/S^1$ is homeomorphic to $\mathbb CP^n$ under a given identification
negative_passages.docid: 131289
negative_passages.text: Polynomial with a root modulo every prime but not in $\mathbb{Q}$.
negative_passages.docid: 108550
negative_passages.text: Elements of $\mathbb Q(2^{1/3})$
negative_passages.docid: 41035
negative_passages.text: Mapping that proves $\mathbb{Z} \subset \mathbb{Q}$
negative_passages.docid: 46730
negative_passages.text: How to show $\mathbb{Q}(\alpha^{4})=\mathbb{Q}(\alpha)$?
negative_passages.docid: 19230
negative_passages.text: Prime/Maximal Ideals in $\mathbb{Z}[\sqrt d]$