For a commutative ring (if necessary field) $R$, we say a square matrix $X$ over $R $is semisimple iff $X$ is diagonalizable.
On the other hand, R-Module $M$ is semisimple iff $M$ is direct sum of irreducible modules (see https://en.wikipedia.org/wiki/Semisimple_module).
Is there relation each definition?
Moreover I often see the word "semisimple" e.g. "semisimple ring", "semisimple Lie alg." Is it used "semisimple" by same meaning?
According to wikipedia, you call a linear operator $T$ semisimple if every $T$ invariant subspace has a direct summand complement.
If $T$ operates on $V$, I believe this is referring to the module structure on $V$ given by the polynomial ring $k[T]$ on $V$. It means that $V$ becomes a semisimple $k[T]$ module in the sense of module theory.
Note that the wiki article additionally says that it becomes equivalent to $T$ being diagonalizable if $k$ is an algebraically closed field. You can still talk about $V_R$ being a semisimple $R[T]$ module in terms of invariant subspaces having complements, but I don't think you have the link to diagonalizability anymore.
Also note that being a semisimple ring is more stringent: all of $R$'s modules have to be semisimple for $R$ to be a semisimple ring. On the other hand, every ring has some semisimple modules.