Remark 1

By Theorem 1, every left nil ideal of a ring A is contained in

its radical. Let $x$ be a nilpotent and central element of A; then $Ax$ is a nil ideal of A, so $x$ belongs to the radical of A. It is, however, possible that there exist nonzero nilpotent elements of A but that A is without radical. For example, for every integer $n\geqslant 2$, the matrix ring $\mathbf{M}_n(K)$ over a field K is simple, hence without radical (VIII, p. 154, Proposition 4), and it contains nilpotent elements, for example the matrix units $E_{ij}$ with $i\not=j$.