An integral domain whose every nonzero prime ideal contains a nonzero principal ideal that is prime is UFD.
93641
A maximal ideal among those avoiding a multiplicative set is prime
130466
Proving every prime ideal is maximal in a finite dimensional ring over a field.
67699
Integral domain that is integrally closed, one-dimensional and not noetherian
84149
Prove that an infinite ring with finite quotient rings is an integral domain
155991
Is it the case that every quotient ring ${F[x]}/{(p(x))}$ is a PID?
3037
Integral domain without unity has prime characteristic?
111307
If ideal $I$ of domain $R$ is free $R$-module, then $I$ is principal ideal.
116453
Is any quotient of a Euclidean domain by a prime ideal a Euclidean domain?
27167
Ideals in $F[x]$ and Euclidean domains are generated by any element of minimal degree
141659
Every field is an integral domain
74558
Localization of ideals at all primes
74139
Subrings of integral domains have the same identity element.
135586
Prove that the preimage of a prime ideal is also prime.
116000
Prove that prime ideals of a finite ring are maximal
22789
How to determine whether a unique factorization domain is a principal ideal domain?
67203
Complement of maximal multiplicative set is a prime ideal
151018
Proving that any finite integral domain is a field.