query: Show that $\mathbb{Z}[x]/\langle2x-1\rangle$ and $\mathbb{Z}[1/2]$ are isomorphic rings.
positive_passages.docid: 144789
positive_passages.text: identify the ring $\mathbb{Z}[x]/(2x-1)$
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: 86922
negative_passages.text: Ring with 10 elements is isomorphic to $\mathbb{Z}/10 \mathbb{Z} $
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: 18186
negative_passages.text: Prove that ideal $I = \langle X^2,X+1\rangle$ is principal or not (in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$ )
negative_passages.docid: 118073
negative_passages.text: Is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$?
negative_passages.docid: 30753
negative_passages.text: Showing there is no ring whose additive group is isomorphic to $\mathbb{Q}/\mathbb{Z}$
negative_passages.docid: 1720
negative_passages.text: Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$
negative_passages.docid: 116594
negative_passages.text: Finite fields are isomorphic
negative_passages.docid: 19112
negative_passages.text: Ring $R= \mathbb Z_3[x]/\langle x^6-1\rangle$
negative_passages.docid: 120181
negative_passages.text: The ideal generated by $2$ and $ x$ in $\mathbb{Z}[x]$
negative_passages.docid: 41069
negative_passages.text: Justifying $\,\Bbb{Z}[i]/\langle 2\rangle\cong \Bbb{Z}[X]/\langle X^2+1,2\rangle\cong \Bbb{F}_2[X]/\langle X^2+1\rangle$
negative_passages.docid: 90437
negative_passages.text: Let $G = \langle a\rangle$ a cyclic group of order m. Prove that G is isomorphic with $Z_m$.
negative_passages.docid: 33025
negative_passages.text: What is the order of $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ and is it cyclic?
negative_passages.docid: 36137
negative_passages.text: Prove $GL_2(\mathbb{Z}/2\mathbb{Z})$ is isomorphic to $S_3$
negative_passages.docid: 148550
negative_passages.text: Units in quotient ring of $\mathbb Z[X]$
negative_passages.docid: 76100
negative_passages.text: Showing that $\left\langle a,b \mid abab^{-1}\right\rangle \cong \pi_1(K) \cong \left\langle c,d \mid c^2 d^2 \right\rangle$