A regular principal right ideal in a right Noetherian ring is essential
= A regular principal right ideal in a right Noetherian ring is essential
{title2=$c\text{ regular}\implies cR\text{ essential}$}
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 $B+cB+c^2B+\cdots$ an infinite <direct sum> of nonzero <right ideals>. Its finite partial sums contradict the <ascending chain condition>. Thus $cR$ is an <essential right ideal>.