Prove that $I=\{r \in R: f(r) \in J\}$ is a prime ideal in $R$.
135586
Prove that the preimage of a prime ideal is also prime.
50908
Showing a Ring and an Ideal are equal
21946
What do prime ideals in $k[x,y]$ look like?
97740
Homogeneous ideals are contained in homogeneous prime ideals
147871
How do the prime ideals of $\mathbb{Z}_{k}$ look like?
69352
Prove that $I= \{a+bi \in ℤ[i] : a≡b \pmod{2}\}$ is an maximal ideal of $ℤ[i]$.
33245
Show that every ideal of the ring $\mathbb Z$ is principal
121750
Let $R$ be a commutative, unital ring and let $I$ be an ideal of $R.$ Prove that if $I$ is maximal then $R/I$ is a field.
52966
Rings, ideals and units-proof
116943
Proof for maximal ideals in $\mathbb{Z}[x]$
117536
Why is the localization at a prime ideal a local ring?
18186
Prove that ideal $I = \langle X^2,X+1\rangle$ is principal or not (in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$ )
90889
Prime ideals in a finite direct product of rings
56800
Intersection of prime ideals
106550
The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$
50809
Proving $R-S$ contains a prime ideal when $S$ is a multiplicative set
99715
Proper ideal and units proof