Prove that $\mathbb{Z}[i]/\langle a+ib\rangle$ is isomorphic to $\mathbb{Z}/\langle a^2+b^2\rangle$ where $a, b$ are relatively prime
6121
Quotient ring of Gaussian integers
92703
Proving that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic.
94090
How to show that for any abelian group $G$, $\text{Hom}(\mathbb{Z},G)$ is isomorphic to $G$
30753
Showing there is no ring whose additive group is isomorphic to $\mathbb{Q}/\mathbb{Z}$
78118
Algebraically, why is $\mathbb{Z}[i]/(i + 1)$ isomorphic to $\mathbb{Z}_{2}$?
67772
How to find $k$ such that $(\mathbb{Z} \times \mathbb{Z})/ \langle (m,n)\rangle \cong \mathbb{Z} \times \mathbb{Z}_k$
41069
Justifying $\,\Bbb{Z}[i]/\langle 2\rangle\cong \Bbb{Z}[X]/\langle X^2+1,2\rangle\cong \Bbb{F}_2[X]/\langle X^2+1\rangle$
18186
Prove that ideal $I = \langle X^2,X+1\rangle$ is principal or not (in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$ )
40321
$\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$.
135241
Show that $\mathbb{Z}/n^2\mathbb{Z}$ is not isomorphic to other group
344
Prove that if $p\in \mathbb{Z}$ is irreducible, then it is also prime.
26749
$\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$
121280
Proving that $a \mathbb{Z} \cap b\mathbb{Z} = \operatorname{lcm}(a,b)\mathbb Z$
42810
Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?
23879
If $m$ and $n$ are distinct positive integers, then $m\mathbb{Z}$ is not ring-isomorphic to $n\mathbb{Z}$
95764
Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.
105363
$\mathbb Z^n/\langle (a,...,a) \rangle \cong \mathbb Z^{n-1} \oplus \mathbb Z/\langle a \rangle$