query: Is the ideal $\langle x^2,x+1 \rangle$ of $\mathbb{Z}[X]$ a principal ideal ?
positive_passages.docid: 18186
positive_passages.text: Prove that ideal $I = \langle X^2,X+1\rangle$ is principal or not (in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$ )
negative_passages.docid: 13682
negative_passages.text: Is $\langle a,b \mid a^2b^2=1 \rangle$ a semidirect product of $\mathbb{Z}^2$ and $\mathbb{Z}_2$?
negative_passages.docid: 105363
negative_passages.text: $\mathbb Z^n/\langle (a,...,a) \rangle \cong \mathbb Z^{n-1} \oplus \mathbb Z/\langle a \rangle$
negative_passages.docid: 81982
negative_passages.text: Is $\mathbb{Z}[x]$ a principal ideal domain?
negative_passages.docid: 75899
negative_passages.text: Principal ideal domain with finitely many ideals
negative_passages.docid: 106911
negative_passages.text: Square root of $-1$ in the ring $\mathbb{R}[x]/\langle(x^2+1)^2\rangle$
negative_passages.docid: 93929
negative_passages.text: The height of a principal prime ideal
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: 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: 19112
negative_passages.text: Ring $R= \mathbb Z_3[x]/\langle x^6-1\rangle$
negative_passages.docid: 116943
negative_passages.text: Proof for maximal ideals in $\mathbb{Z}[x]$
negative_passages.docid: 76661
negative_passages.text: Checking the maximality of a principal ideal in $R[x]$
negative_passages.docid: 151794
negative_passages.text: Can an ideal in a commutative integral domain be its own square?
negative_passages.docid: 78143
negative_passages.text: Proving $\mathbb{F}_p/\langle f(x)\rangle$ with $f(x)$ irreducible of degree $n$ is a field with $p^n$ elements
negative_passages.docid: 113152
negative_passages.text: Prove Principal Ideal Domain from Bezout's condition, and terminating divisibility chain
negative_passages.docid: 99198
negative_passages.text: Principal ideal and free module
negative_passages.docid: 102721
negative_passages.text: Number of elements in the ring $\mathbb Z [i]/\langle 2+2i\rangle$