Can any one suggest me an example of a ring $R$ and an $R$-module $A$ s.t. torsion of module $A$ is not a sub-module?
Torsion of module $A$, i.e. $\operatorname{Tor}(A)$, denotes all torsion elements in module $A$.
*(And we already know that if $R$ is a commutative integral domain then $\operatorname{Tor}(A)$ is a submodule of $A$).
Best Answer
An example for a commutative ring which is not a domain: $R=\{0,a,1-a,1\}$ with $a^2=a,a+a=1+1=0$ and $A=R$. The elements $a,1-a$ are torsion, but $a+(1-a)=1$ isn't.