An example for two finitely generated and residually finite whose quotients are finite and Isomorphic, but themselves are not Isomorphic
59902
Example of $2$ non-isomorphic groups that have the same quotients
124864
A not free $\mathbb{Z}$-module
92703
Proving that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic.
4624
Example of a finitely generated module with submodules that are not finitely generated
95764
Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.
122353
Status of the classification of non-finitely generated abelian groups.
96892
Number of Non-isomorphic models of Set Theory
79056
Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$
37904
Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.
115129
Constructing an explicit isomorphism between finite extensions of finite fields
80879
Can a quotient field ever be finitely generated as an algebra?
70872
Isomorphism of quotient modules implies isomorphism of submodules?
156679
$K[X,Y]/(XY-1)$ not isomorphic to $K[T]$
2254
Finite group with isomorphic normal subgroups and non-isomorphic quotients?
17978
Isomorphism classes of $\mathbb{Z}[i]$ modules.
103427
When is $\operatorname{Hom}_A(M,N)$ finitely generated?
135241
Show that $\mathbb{Z}/n^2\mathbb{Z}$ is not isomorphic to other group