Noetherian domain R is a UFD if every prime ideal of height 1 in R is principal.
6507
Every prime ideal of height 1 in a UFD is principal
81743
Prime ideals in $C[0,1]$
74370
$P$ is a prime ideal of $R$ iff $R/P$ is an integral domain. : $P≠R$
149753
The structure of a Noetherian ring in which every element is an idempotent.
84663
$A$ local Noetherian ring with principal maximal ideal implies PIR?
24224
How to prove the ring of Laurent polynomials over a field is a principal ideal domain?
93658
Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible
138882
A prime poset of ideals
94157
Normal Ring and Prime Ideal whose Square is Principal
155991
Is it the case that every quotient ring ${F[x]}/{(p(x))}$ is a PID?
84386
Domain strictly contained in the intersection of localizations at the primes of height one
51578
Noetherian ring with finitely many height $n$ primes
113152
Prove Principal Ideal Domain from Bezout's condition, and terminating divisibility chain
43103
Unique factorization domain and principal ideals
152109
In a UFD if $x$ is irreducible, then $(x)$ is prime
50809
Proving $R-S$ contains a prime ideal when $S$ is a multiplicative set
121986
Do the Laurent polynomials over $\mathbb{Z}$ form a principal ideal domain?