Examples for a K-module with K ring, but not an integral domain?
136917
Torsion elements do not form a submodule.
46047
Check whether the quotient ring $R:=\mathbb{C}[x,y,z]/(xy-z^2)$ is a field or an integral domain
83812
Artinian ring and faithful module of finite length
147722
Is there a nice example of integral domain $R$ and $s,t \in R$, with $Rs \cap Rt$ not principal but lcm($s,t$) exists?
47114
What is necessary and/or sufficient for polynomials to provide isomorphic quotient rings?
86724
Example of a non-zero module that has no associated primes
33576
Non-principal ideal in $K[x,y]$?
153852
Example of modules that are projective but not free; torsion-free but not free
25764
Example of a ring such that $R^2\simeq R^3$, but $R\not\simeq R^2$ (as $R$-modules)
26368
Finitely generated graded modules over $K[x]$
75641
Is there a (f.g., free) module isomorphic to a quotient of itself?
3037
Integral domain without unity has prime characteristic?
46258
Example of Artinian module that is not Noetherian
92420
Characteristic of  a Non-unital Integral Ring
42652
Must $k$-subalgebra of $k[x]$ be finitely generated?
30465
Determining which rings have a module which is not semisimple
112099
Polynomial ring with integral coefficients is integral