Annihilators of nilpotent elements

Abstract

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)=cardI. In particular, cardA(x)=cardR; and this is applied to prove that if N is the set of nilpotent elements of R and R≠N, then card(R\N)≥cardN.