query: Is the ideal generated by the set $\{2,x\}$ in $\mathbb Z[x]$ a Principal Ideal?
positive_passages.docid: 1720
positive_passages.text: Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$
negative_passages.docid: 138114
negative_passages.text: Proving the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain
negative_passages.docid: 112043
negative_passages.text: Show that the ideal $ (2, 1 + \sqrt{-7} ) $ in $ \mathbb{Z} [\sqrt{-7} ] $ is not principal
negative_passages.docid: 113152
negative_passages.text: Prove Principal Ideal Domain from Bezout's condition, and terminating divisibility chain
negative_passages.docid: 44083
negative_passages.text: prove that maximal ideal in $\mathbb{Z}$ generated by a prime number
negative_passages.docid: 154845
negative_passages.text: Irreducible elements in  $\mathbb{Z}[\sqrt{-2}]$ and is it a Euclidean domain?
negative_passages.docid: 129302
negative_passages.text: Proving whether ideals are prime in $\mathbb{Z}[\sqrt{-5}]$
negative_passages.docid: 42170
negative_passages.text: $R$ integral domain where every prime ideal is principal then set of non-principal ideals has maximal element
negative_passages.docid: 80397
negative_passages.text: Show that this set of polynomials is ideal in F[x]
negative_passages.docid: 149747
negative_passages.text: Prove that if I is a prime ideal of R then I[x] is a prime ideal of R[x]
negative_passages.docid: 147871
negative_passages.text: How do the prime ideals of $\mathbb{Z}_{k}$ look like?
negative_passages.docid: 41629
negative_passages.text: Structure of ideals in the product of two rings
negative_passages.docid: 21946
negative_passages.text: What do prime ideals in $k[x,y]$ look like?
negative_passages.docid: 70047
negative_passages.text: Is the ideal generated by irreducible element in principal ideal domain maximal?
negative_passages.docid: 144789
negative_passages.text: identify the ring $\mathbb{Z}[x]/(2x-1)$
negative_passages.docid: 60674
negative_passages.text: Finitely generated idempotent ideals are principal: proof without using Nakayama's lemma
negative_passages.docid: 18186
negative_passages.text: Prove that ideal $I = \langle X^2,X+1\rangle$ is principal or not (in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$ )