 A regular principal right ideal in a right Noetherian ring is essential

If $c$ is a regular element of a ring in a right Noetherian ring and a nonzero right ideal $B$ had $B\cap cR=0$, cancellation would make $B+cB+c^{2}B+\cdots$ an infinite direct sum of nonzero right ideals. Its finite partial sums contradict the ascending chain condition. Thus $cR$ is an essential right ideal.