Right fully idempotent rings need not be left fully idempotent

Abstract

All rings in this paper are associative but not necessarily with an identity. The ring R with an identity adjoined will be denoted by R#.To denote that I is an ideal (right ideal, left ideal) of a ring R we write I ◃ R (I <rR, I <1R).A ring R is called right (left) fully idempotent if for every I <rR (I<1R), I = I2.At the conference “Methoden der Modul und Ringtheorie” in Oberwolfach, Germany in 1993, J. Clark raised the question as to whether every right fully idempotent ring is left fully idempotent (see also [3]). A similar question was raised by S. S. Page in [5]. In this note we answer the questions in the negative.We start with some general observations most of which are perhaps well known. We include their simple proofs for completeness.