query: Is $(2, X)$ a prime ideal in $\mathbb{Z}[X]$?
positive_passages.docid: 37929
positive_passages.text: Classification of prime ideals of $\mathbb{Z}[X]$
negative_passages.docid: 137858
negative_passages.text: Showing that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal, and also finding its cardinality?
negative_passages.docid: 92519
negative_passages.text: ideal behind Maximal ideals are Prime ideals
negative_passages.docid: 135586
negative_passages.text: Prove that the preimage of a prime ideal is also prime.
negative_passages.docid: 99423
negative_passages.text: Is it true every ideal of $\mathbb{Z}[x]$ is finitely generated?
negative_passages.docid: 62917
negative_passages.text: Associated Prime Ideals in a Noetherian Ring; Exercise 6.4 in Matsumura
negative_passages.docid: 110022
negative_passages.text: Showing prime and maximal ideal
negative_passages.docid: 42875
negative_passages.text: The prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $.
negative_passages.docid: 149203
negative_passages.text: Symbolic power of a prime ideal is primary
negative_passages.docid: 136654
negative_passages.text: $(x_1-a_1, x_2-a_2)$ is a maximal ideal of $K[x_1,x_2]$
negative_passages.docid: 93641
negative_passages.text: A maximal ideal among those avoiding a multiplicative set is prime
negative_passages.docid: 116430
negative_passages.text: If $I = \langle 2\rangle$, why is $I[x]$ not a maximal ideal of $\mathbb Z[x]$, even though $I$ is a maximal ideal of $\mathbb Z$?
negative_passages.docid: 104486
negative_passages.text: Ideal in $\mathbb Z[x]$ which is not two-generated
negative_passages.docid: 40975
negative_passages.text: Quotient of the ring of integers of a number field by a prime ideal
negative_passages.docid: 50809
negative_passages.text: Proving $R-S$ contains a prime ideal when $S$ is a multiplicative set
negative_passages.docid: 70856
negative_passages.text: Construct ideals in $\mathbb Z[x]$ with a given least number of generators
negative_passages.docid: 72230
negative_passages.text: Equivalence of definitions of prime ideal in commutative ring