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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.

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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.

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  • $\begingroup$ You can also define a $\mid_B$ with $a\mid_B b$ if $b=c_1ac_2$ for some $c_1,c_2$. This is just the transitive completion of $\mid_{\ell}\cup\mid_{r}$. $\endgroup$ – Thomas Andrews Jul 20 '17 at 16:32
  • $\begingroup$ @ThomasAndrews Interesting... I had not seen that one before. Another iteration of that might be to say $a|_Ib$ if $(b)\subseteq (a)$ (two-sided principal ideals), to play with the containment-divisibility connection. $\endgroup$ – rschwieb Jul 20 '17 at 16:35
  • $\begingroup$ It means we can talk about divisibility of matrices too as set of square matrices form a non-commutative ring under usual matrix addition and multiplication, using the definition of divisibility as," we say in a ring R an element a is said to divide an element b in if there is an element c in R such that b=ac". $\endgroup$ – Rattan verma Jul 21 '17 at 5:12
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I got the example. X^2 + ix - jx + k = (x - j)(x + i), but X^2 + ix - jx + k = (x + i)(x - j) + 2k in Hamiltonian Quaternions ring.

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