query: Are quotients of a PID by non-prime ideals ever a PID?
positive_passages.docid: 145879
positive_passages.text: $R$ is PID, so $R/I$ is PID, and application on $\mathbb{Z}$ and $\mathbb{N}$
negative_passages.docid: 135152
negative_passages.text: Non-UFD integral domain such that prime is equivalent to irreducible?
negative_passages.docid: 27916
negative_passages.text: Are the determinantal ideals prime?
negative_passages.docid: 102282
negative_passages.text: Let $R$ be a PID, and let $\pi\in R$ be prime. Is it $R/\langle\pi\rangle \times R/\langle \pi\rangle$ a cyclic $R$-module?
negative_passages.docid: 28987
negative_passages.text: One-dimensional [Noetherian] UFD is a PID
negative_passages.docid: 129302
negative_passages.text: Proving whether ideals are prime in $\mathbb{Z}[\sqrt{-5}]$
negative_passages.docid: 93658
negative_passages.text: Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible
negative_passages.docid: 97740
negative_passages.text: Homogeneous ideals are contained in homogeneous prime ideals
negative_passages.docid: 143140
negative_passages.text: When are quotients of homeomorphic spaces homeomorphic?
negative_passages.docid: 84663
negative_passages.text: $A$ local Noetherian ring with principal maximal ideal implies PIR?
negative_passages.docid: 136855
negative_passages.text: Proof that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ is a PID
negative_passages.docid: 13901
negative_passages.text: Finitely many prime ideals lying over $\mathfrak{p}$
negative_passages.docid: 121986
negative_passages.text: Do the Laurent polynomials over $\mathbb{Z}$ form a principal ideal domain?
negative_passages.docid: 85915
negative_passages.text: Principal prime ideals are minimal among prime ideals in a UFD
negative_passages.docid: 16984
negative_passages.text: Finite extension of domains : preimage of non-zero ideal is non-zero?
negative_passages.docid: 27167
negative_passages.text: Ideals in $F[x]$ and Euclidean domains are generated by any element of minimal degree
negative_passages.docid: 149412
negative_passages.text: Every prime ideal of a finitely generated $\mathbb{R}$-algebra is an intersection of maximal ideals?