Left Noetherian domain Ore lemma ($R\text{ left Noetherian domain}⟹R\setminus \{0\}\text{ left Ore}$)

In a noncommutative domain that is a left Noetherian ring, two nonzero principal left ideals always intersect nontrivially. If $Ra\cap Rb=0$, a relation ${\sum }_{i=0}^m{u}_{i}a{b}^i=0$ first forces $u_{0}a\in Ra\cap Rb$, then $u_0=0$. Right cancellation of $b$ and repetition give all $u_i=0$. Hence ${\sum }_{i\ge 0}Ra{b}^i$ is an infinite direct sum of nonzero left ideals, whose finite partial sums violate the ascending chain condition. A nonzero member of $Rr\cap Rs$ supplies the equality $s^{\prime }r=r^{\prime }s$ with $s^{\prime }\ne 0$, proving the left Ore set condition for all nonzero elements.