Modules

$(x + (x,y)^2)$ over the ring $F_2[x,y]/(x,y)^2$

$(x)/(x^2)$ over the ring $\mathbb R[x]/(x^2)$

$2$-adic integers: $\mathbb Z_2$ over the ring $2$-adic integers: $\mathbb Z_2$

$\bigoplus_{i=0}^\infty \mathbb Q$ over the opposite ring of Michler & Villamayor's right-not-left V ring

$\bigoplus_{i=1}^\infty \mathbb Z$ over the ring $\mathbb Z$: the ring of integers

$\bigoplus_{i=1}^\infty \mathbb Z_{(2)}$ over the ring $\mathbb Z_{(2)}$

$\bigoplus_{i=1}^\infty F_2$ over the ring $\prod_{i=1}^\infty F_2$

$\mathbb Q$ over the ring $\mathbb Z$: the ring of integers

$\mathbb Q/\mathbb Z$ over the ring $\mathbb Z$: the ring of integers

$\mathbb Q\times \frac{\mathbb Z}{p\mathbb Z}$, where $p$ is prime over the ring $\mathbb Z$: the ring of integers

$\mathbb Q^n$ over the ring $M_n(\mathbb Q)$

$\mathbb R/\mathbb Z$: the circle group over the ring $\mathbb Z$: the ring of integers

$\mathbb R[x_1,x_2,x_3,\ldots]$ over the ring $\mathbb R[x_1, x_2,x_3,\ldots]$

$\mathbb Z$ over the ring $\mathbb Z$: the ring of integers

$\mathbb Z\times \frac{\mathbb Z}{p\mathbb Z}$, $p$ prime over the ring $\mathbb Z$: the ring of integers

$M_n(\mathbb Q)$ over the ring $M_n(\mathbb Q)$

$T_n(F_2)$ over the ring $T_n(F_2)$

$Z(p^\infty)$: the Prüfer $p$ group over the ring $\mathbb Z$: the ring of integers

A 2-generated faithful torsion module over the ring $F_2[x,y]/(x,y)^2$

Finitely cogenerated, not Artinian over the ring Finitely cogenerated, not semilocal ring

Indecomposable, not uniform module over the ring $F_2[x,y]/(x,y)^2$

Interval monoid ring (right regular module) over the ring Interval monoid ring

Uniserial, not endolocal module over the ring 2-dimensional uniserial domain