Let $R$ be a non-commutative integral domain with unity which is also a right Noetherian ring. By integral domain I mean that the product of nonzero elements is always nonzero. I am trying to show the following easy thing: Let $a,b\in R$ be two nonzero elements.
I want to show that there exists $p,q \in R$ such that $aq=bp\neq 0$
I am searching for a down to earth proof. (not by any famous theorem) Thanks in advance for any help.