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 $A$).
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.
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.