For a non-zero principal ideal $I=(x)$ of a ring of integers of an algebraic number field, $|A/I|=| N_{L|\mathbb Q } (x)|$
134939
Norm of element $\alpha$ equal to absolute norm of principal ideal $(\alpha)$
137858
Showing that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal, and also finding its cardinality?
33576
Non-principal ideal in $K[x,y]$?
138114
Proving the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain
89271
If $F$ is a field, then $F[x]$ is a principal ideal domain proof question
147871
How do the prime ideals of $\mathbb{Z}_{k}$ look like?
146801
Primary decomposition of an ideal
133975
Localising a polynomial ring and non-maximal prime ideal
148788
Polynomial ring ideal. Show that $(f) \cap R_N = (0) \iff f \notin R_N$
151786
Proving that there is no ideal $I$ in $\mathbb{Z}[t]$ such that $\mathbb{Z}[t]/I \cong \mathbb{Q}$
145594
Splitting of prime ideals in algebraic extensions
148208
In Noetherian ring $R$, ideal $I\subset R$, $P\in\mathrm{Ass}(I),P=I:a$ for some $a\in R$.
464
Prove that the ideal $(X_1-a_1,...,X_n-a_n)$ is maximal in $K[X_1,\dots,X_n]$
93929
The height of a principal prime ideal
1720
Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$
102341
Primary decomposition of a monomial ideal
62917
Associated Prime Ideals in a Noetherian Ring; Exercise 6.4 in Matsumura