query: Subring of rationals is a 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: 134840
negative_passages.text: Generators and relations for $\text{PSL}(2, \mathbb{F}_q)$
negative_passages.docid: 22245
negative_passages.text: Show the union of two subrings is generally not a subring
negative_passages.docid: 75899
negative_passages.text: Principal ideal domain with finitely many ideals
negative_passages.docid: 76202
negative_passages.text: Sufficient conditions for being a PID
negative_passages.docid: 122487
negative_passages.text: Degree of Rational Function
negative_passages.docid: 107089
negative_passages.text: finding rational roots of polynomials
negative_passages.docid: 145879
negative_passages.text: $R$ is PID, so $R/I$ is PID, and application on $\mathbb{Z}$ and $\mathbb{N}$
negative_passages.docid: 119360
negative_passages.text: Find all subrings of $\mathbb{Z}^2$
negative_passages.docid: 136770
negative_passages.text: Any field that contains the integers contains the rationals as a subfield (Analysis proof)
negative_passages.docid: 62134
negative_passages.text: Show that if $ R $ is an integral domain then a polynomial in $ R[X] $ of degree $ d $ can have at most $ d $ roots
negative_passages.docid: 150170
negative_passages.text: Subrings of fraction fields
negative_passages.docid: 141477
negative_passages.text: Decomposition of finitely generated graded modules over PID
negative_passages.docid: 48804
negative_passages.text: Enumeration of rationals from Stein-Shakarchi's Real Analysis (Chapter 1, Exercise 24)
negative_passages.docid: 157420
negative_passages.text: Question about Subrings and integraly closed
negative_passages.docid: 132931
negative_passages.text: Determining the parity of the coefficients of a quadratic given that there is a rational solution
negative_passages.docid: 126513
negative_passages.text: Automorphism of the group of rational numbers under addition