Example when the torsion of a module is not a submodule. 
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 $A$).
 A: 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.
A: Let me propose a more elementary example.
Let $R$ be $(\mathbb Z_6,+,\times)$. Let $A=R$.
Then $A$ is a $R$-module.
Let Tor$(A):=\{a\in A|ra=0 \mbox{ for some nonzero }r\in R\}$.
Note that $\bar{1}$ and $\bar{5}$ are the inverses of each other. So they are not in Tor$(A)$.
Note that $\bar{2}\,\bar{3}=\bar{3}\,\bar{2}=\bar{0}$ and $\bar{3}\,\bar{4}=\bar{0}$. So, $\bar{2}, \bar{3},\bar{4}\in$ Tor$(A)$. Also, $\bar{0}\in$ Tor$(A)$ is trivial.
Now we easily see that $\bar{2}+\bar{3}=\bar{5}\notin$ Tor$(A)$. Hence, Tor$(A)$ is not a $R$-submodule of $A$.
A: actually, the Torsion subset is a submodule for all $R$-modules $M$.
Recall that $m\in M$ is called torsion, if there is $r\in R$ which is regular (i.e. not a zero divisor) such that $r.m=0$.
Assume $m$ and $m'$ are torsion with corresponding regular elements $r$ and $r'$. Then $rr'$ is non-zero and still not a zero divisor and we have $rr'.(m+m')=0$. Hence $m+m'$ is torsion.
It is trivial to see that if $m$ is torsion then the same is true for $s.m$ for all $s\in R$.
Note also that over a finite ring every module is torsion.
To see this, notice that an element of a finite ring is either a unit or a zero-divisor. Hence all regular elements act invertibly on modules and hence cannot kill any non-zero elements.
