query: $\mathbb{Z} \times \mathbb{Z} $ is a PID or not?
positive_passages.docid: 149783
positive_passages.text: $\mathbb Z\times\mathbb Z$ is principal but is not a PID
negative_passages.docid: 62566
negative_passages.text: Does $X\times S^1\cong Y\times S^1$ imply that $X\times\mathbb R\cong Y\times\mathbb R$?
negative_passages.docid: 121845
negative_passages.text: Quotient of polynomials, PID but not Euclidean domain?
negative_passages.docid: 14447
negative_passages.text: Direct proof that $\pi$ is not constructible
negative_passages.docid: 11259
negative_passages.text: Prove $R\times S$ is not an integral domain
negative_passages.docid: 28987
negative_passages.text: One-dimensional [Noetherian] UFD is a PID
negative_passages.docid: 120035
negative_passages.text: Pattern in Isomorphisms between $\mathbb Z_m \times \mathbb Z_n$ and $\mathbb Z_{mn}$
negative_passages.docid: 51687
negative_passages.text: Is the set $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \mathbb{Q}$ non-empty?
negative_passages.docid: 132193
negative_passages.text: Why $\Bbb Z$ = (integers) is an integral domain?
negative_passages.docid: 47522
negative_passages.text: group generators of $(\mathbb Z_{17}-\{0\},\times)$
negative_passages.docid: 31391
negative_passages.text: bijection between $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$
negative_passages.docid: 153664
negative_passages.text: When is $\Bbb{Z[\zeta_n]}$ a PID?
negative_passages.docid: 6161
negative_passages.text: The cartesian product $\mathbb{N} \times \mathbb{N}$ is countable
negative_passages.docid: 114394
negative_passages.text: Determine whether $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ are isomorphic groups or not.
negative_passages.docid: 31451
negative_passages.text: Proving a subring of $\mathbb{Q}$ containing $\mathbb{Z}$ is a PID
negative_passages.docid: 149605
negative_passages.text: Homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$?
negative_passages.docid: 81238
negative_passages.text: Is $O(2)$ really not isomorphic to $SO(2)\times \{-1,1\}$?