I was reading " Topics in Algebra" by I.N. Herstein and in this book divisibility is defined on a commutative ring. So I just want to know if there is a concept on non-commutative rings of divisibility.
You can, but you'd have to specify sides. What I mean is that "$a$ divides $b$ on the right" may not mean the same thing as "$a$ divides $B$ on the left."
You could say $a|_r b$ if $a=cb$ for some $c$, and $a|_\ell b$ if $a=bc$ for some $c$. This would correspond to containment between principal left ideals and containment between principal right ideals.
A good example is trying to factor polynomials in $\mathbb H[x]$. I don't have an example at hand, but I'm pretty sure I've seen an example of such a polynomial that was divisible on the left by a linear factor $x-\alpha$ but not divisible on the right by $x-\alpha$.
Another way to get an example of interesting things happening is if you took the free algebra $\mathbb Q\langle x,y,z\rangle$ modulo the ideal containing $xy-z$, so that $x$ divides $z$ on the left, but not on the right.