Annihilators of nilpotent elements. (Q2368475)

Author's summary: 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 \(\text{card}(A(x)\cap I)=\text{card\,}I\). In particular, \(\text{card\,}A(x)=\text{card\,}R\); and this is applied to prove that if \(N\) is the set of nilpotent elements of \(R\) and \(R\neq N\), then \(\text{card}(R\setminus N)\geq\text{card\,}N\).