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Right fully idempotent rings need not be left fully idempotent

Published online by Cambridge University Press:  18 May 2009

R. R. Andruszkiewicz  and
E. R. Puczyłowski
R. R. Andruszkiewicz
Affiliation:
Institute of Mathematics, University of Warsaw, Białystok Division, Akademicka 2, 15–267 Białystok, Poland
E. R. Puczyłowski
Affiliation:
Institute of Mathematics, University of Warsaw, Banacha, 02-097 Warsaw, Poland
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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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 37 , Issue 2 , May 1995 , pp. 155 - 157
Copyright
Copyright © Glasgow Mathematical Journal Trust 1995

References

REFERENCES

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