International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 21, Pages 3517-3519

Annihilators of nilpotent elements

Abraham A. Klein

Department of Pure Mathematics, School of Mathematical Sciences, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Received 23 February 2005; Revised 4 September 2005

Copyright © 2005 Abraham A. Klein. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 RN, then card(R\N)cardN.