A Noetherian domain $R$ is a UFD iff $I_{fg}=\{h\in R \mid hg\in(f)\}$ is principal for all $f,g\in R$.
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A Noetherian integral domain is a UFD iff $(f):(g)$ is principal
74314
Noetherian ring and the same prime divisors
6507
Every prime ideal of height 1 in a UFD is principal
42170
$R$ integral domain where every prime ideal is principal then set of non-principal ideals has maximal element
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Associated Prime Ideals in a Noetherian Ring; Exercise 6.4 in Matsumura
154495
Prove that a nontrivial subring of a ring that is a domain has the same identity
151018
Proving that any finite integral domain is a field.
84663
$A$ local Noetherian ring with principal maximal ideal implies PIR?
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Let $R$ be a commutative ring. If $R[X]$ is a principal ideal domain, then $R$ is a field.
53367
In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?
4196
A valid proof for the invariance of domain theorem?
7645
Noetherian property for exact sequence
85206
Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.
87103
Quick question about ideals of field $F$ in $F[x]$ are principal.
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Is every Noetherian module finitely generated?
138114
Proving the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain
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Ideals in $F[x]$ and Euclidean domains are generated by any element of minimal degree