Essential right ideal ($I\cap B\ne 0\text{ for every nonzero right ideal }B$)

An essential right ideal is an essential submodule of the right regular module of a ring. Every right ideal containing an essential right ideal 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.