Let x be a nilpotent element of an infinite ring R (not necessarily with 1 ). We prove that A ( x ) —the two-sided annihilator of x —has a large intersection with any infinite ideal I of R in the sense that card ( A ( x ) ∩ I ) = card I . In particular, card A ( x ) = card R ; and this is applied to prove that if N is the set of nilpotent elements of R and R ≠ N , then card ( R \ N ) ≥ card N .