query: What's the meaning of factors 'collapsing' in quotients like $\mathbb{Z}_4\times\mathbb{Z}_6/\langle(0,1)\rangle$?
positive_passages.docid: 147968
positive_passages.text: Calculate the Factor Group $(\mathbb{Z}_4 \times \mathbb{Z}_6)/\langle(0,2)\rangle$
negative_passages.docid: 10589
negative_passages.text: How to factorize $5$ in $\mathbb{Z}[\root 3 \of 2]$?
negative_passages.docid: 137285
negative_passages.text: Prove $\mathbb{Z}_2 \times \mathbb{Z}_2$ is not isomorphic to $\mathbb{Z}_4$
negative_passages.docid: 18198
negative_passages.text: Lang Algebra Chapter 6 Theorem 9.4
negative_passages.docid: 137949
negative_passages.text: If the quotient group $G/N$ be isomorphic to $G_2$, is $G$ isomorphic to $G_1\times G_2$ where $N$ is isomorphic to $G_1$?
negative_passages.docid: 19112
negative_passages.text: Ring $R= \mathbb Z_3[x]/\langle x^6-1\rangle$
negative_passages.docid: 141258
negative_passages.text: Why does $M \mathbin{\otimes_R} N \cong M_\mathfrak{p} \mathbin{\otimes_{R_\mathfrak{p}}} N$?
negative_passages.docid: 60268
negative_passages.text: $\textit{Why}$ the cases in 4 $\nmid n^2 - 3?$ (bases cases in modular induction)
negative_passages.docid: 30172
negative_passages.text: Why is the order of $\mathbb{Z}[i]/\langle 3+i\rangle$ equal to 10?
negative_passages.docid: 133384
negative_passages.text: Question regarding to canonical factorization of $n!$?
negative_passages.docid: 93291
negative_passages.text: Method for determining irreducibles and factorising in $\mathbb Z[\sqrt{d}]$
negative_passages.docid: 9794
negative_passages.text: Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime
negative_passages.docid: 114916
negative_passages.text: $\mathbb Z_{mn}$ isomorphic to $\mathbb Z_m\times\mathbb Z_n$ whenever $m$ and $n$ are coprime
negative_passages.docid: 121404
negative_passages.text: How to factor polynomials in $\mathbb{Z}_n$?
negative_passages.docid: 26749
negative_passages.text: $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$
negative_passages.docid: 43230
negative_passages.text: Is $\langle1-i\rangle$ a maximal ideal in $\mathbb{Z}[i]$?
negative_passages.docid: 138058
negative_passages.text: Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.