What does the following sentence mean?
"Let $a$ in a commutative ring with unity $R$ such that $a$ is admitting power with torsion of $\mathbb{Z}$-module".
Does this mean there exist positive integers $n,t$ such that $ta^n=0$. if yes, then does this implies $na^n=0$.
During my read of the proof of some lemma, I found "Suppose that a power of $a$ is with torsion. We can find an integer $r$ such that $ra^r=0$"
It confused me, why $ra^r=0$, is not suppose to be $sa^r=0$ for some integer $s$?
Yes, your idea is right: the condition means there are positive integers $t$ and $n$ such that $ta^n=0$.
Take $r=tn$; then $ra^r=n(ta^n)a^t=0$.