$\mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}$ are not equivalent as abelian group models.
128989
Is ${\bf Z} \times {\bf Z}$ elementarily equivalent to ${\bf Z}$?
128732
$\mathbb Z_{p^n}$ is not the direct product of any family of its proper subgroups
114916
$\mathbb Z_{mn}$ isomorphic to $\mathbb Z_m\times\mathbb Z_n$ whenever $m$ and $n$ are coprime
104486
Ideal in $\mathbb Z[x]$ which is not two-generated
127978
Another point of view that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is cyclic.
78118
Algebraically, why is $\mathbb{Z}[i]/(i + 1)$ isomorphic to $\mathbb{Z}_{2}$?
118299
Show that $(\mathbb{Z}[x]/(x^{n+1}))^{\times}\cong \mathbb{Z}/2\mathbb{Z}\times\Pi_{i=1}^n\mathbb{Z}$
128652
Subgroups of $\mathbb Z \times(\mathbb Z/n\mathbb Z)$
126927
$\mathbb Q$ is not the direct product of two non-trivial groups.
18182
An abelian group $G$ with ${\rm Aut}(G)$ non-abelian
124455
Why multiplicative group $\mathbb{Z}_n^*$ is not cyclic for $n = 2^k$ and $k \ge 3$
68672
Commutator subgroup of rank-2 free group is not finitely generated.
122353
Status of the classification of non-finitely generated abelian groups.
12110
How are $2\mathbb{Z}\ncong3\mathbb{Z}$ different as rings?
1720
Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$
75970
$\mathbb Z[1/2]$ is not finitely generated?
86234
Explain why $\mathbb{Z \times Z}$ and $\mathbb{R \times R}$ is not a field