query: Show $\mathbb{R}[x]/(x^2)$ is a PID
positive_passages.docid: 155991
positive_passages.text: Is it the case that every quotient ring ${F[x]}/{(p(x))}$ is a PID?
negative_passages.docid: 40549
negative_passages.text: How to proof the the derivative of $x^2$ is $2x$?
negative_passages.docid: 143831
negative_passages.text: Showing $\mathbb{Q}$ is homeomorphic to $\mathbb{Q}^2$
negative_passages.docid: 23721
negative_passages.text: Prove that the Gaussian Integer's ring is a Euclidean domain
negative_passages.docid: 19424
negative_passages.text: Showing $(M^\perp)^\perp=\overline{M}$
negative_passages.docid: 62755
negative_passages.text: The simplest way of proving that $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}| = c$
negative_passages.docid: 109295
negative_passages.text: Proving $1-\exp(-4x^2/ \pi) \ge \text{erf}(x)^2$
negative_passages.docid: 53575
negative_passages.text: Describe $R=\mathbb{Z}[X]/(X^2-3,2X+4)$
negative_passages.docid: 59424
negative_passages.text: Showing that $B=\{(x,y) \in \Bbb{R^2} \mid \ xy>0 \}$ is an open set
negative_passages.docid: 104548
negative_passages.text: Geometric proof for irrationality of $\pi$
negative_passages.docid: 118299
negative_passages.text: Show that $(\mathbb{Z}[x]/(x^{n+1}))^{\times}\cong \mathbb{Z}/2\mathbb{Z}\times\Pi_{i=1}^n\mathbb{Z}$
negative_passages.docid: 47975
negative_passages.text: Closed cone in the euclidean space $\mathbb{R}^n$
negative_passages.docid: 41872
negative_passages.text: Show that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus\{(0,0)\}$
negative_passages.docid: 1020
negative_passages.text: Proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain
negative_passages.docid: 121845
negative_passages.text: Quotient of polynomials, PID but not Euclidean domain?
negative_passages.docid: 76202
negative_passages.text: Sufficient conditions for being a PID
negative_passages.docid: 28987
negative_passages.text: One-dimensional [Noetherian] UFD is a PID