query: When is a Unique Factorization Domain a Principal Ideal domain
positive_passages.docid: 28987
positive_passages.text: One-dimensional [Noetherian] UFD is a PID
negative_passages.docid: 78380
negative_passages.text: Dedekind domain with a finite number of prime ideals is principal
negative_passages.docid: 113136
negative_passages.text: Generators of the ideal correspond to $d$-uple embedding
negative_passages.docid: 87151
negative_passages.text: Primary ideals in Noetherian rings
negative_passages.docid: 74139
negative_passages.text: Subrings of integral domains have the same identity element.
negative_passages.docid: 109688
negative_passages.text: Zero-dimensional ideals and finite-dimensional algebras
negative_passages.docid: 123024
negative_passages.text: A Noetherian integral domain is a UFD iff $(f):(g)$ is principal
negative_passages.docid: 42170
negative_passages.text: $R$ integral domain where every prime ideal is principal then set of non-principal ideals has maximal element
negative_passages.docid: 97902
negative_passages.text: In a integral domain every prime element is irreducible
negative_passages.docid: 99198
negative_passages.text: Principal ideal and free module
negative_passages.docid: 31049
negative_passages.text: If $p$ is an irreducible element of an integral domain $D$, and if $e$ belongs to $D^\ast$ prove that $ep$ is also ireducible
negative_passages.docid: 78626
negative_passages.text: Let $R$ be a commutative ring. If $R[X]$ is a principal ideal domain, then $R$ is a field.
negative_passages.docid: 102341
negative_passages.text: Primary decomposition of a monomial ideal
negative_passages.docid: 146978
negative_passages.text: Minimal and characteristic polynomial have same set of irreducible factors
negative_passages.docid: 84663
negative_passages.text: $A$ local Noetherian ring with principal maximal ideal implies PIR?
negative_passages.docid: 512
negative_passages.text: Finding generator for the ideal generated by $a$ and $b$ in an euclidean domain
negative_passages.docid: 80679
negative_passages.text: Principal ideal rings that are not integral domains