Quotient of a PID by a prime ideal is a PID too.
138114
Proving the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain
154239
Can a prime in a Dedekind domain be contained in the union of the other prime ideals?
152109
In a UFD if $x$ is irreducible, then $(x)$ is prime
102282
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?
62917
Associated Prime Ideals in a Noetherian Ring; Exercise 6.4 in Matsumura
90889
Prime ideals in a finite direct product of rings
81743
Prime ideals in $C[0,1]$
74558
Localization of ideals at all primes
157967
Contraction of quotient ideal is quotient of contractions?
99715
Proper ideal and units proof
119056
If in a UFD every maximal ideal is principal then it is a PID
63659
If D is an Integral Domain and has finite characteristic p, prove p is prime.
97740
Homogeneous ideals are contained in homogeneous prime ideals
149747
Prove that if I is a prime ideal of R then I[x] is a prime ideal of R[x]
6507
Every prime ideal of height 1 in a UFD is principal
110022
Showing prime and maximal ideal
31451
Proving a subring of $\mathbb{Q}$ containing $\mathbb{Z}$ is a PID