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A Look at the FaithConjecture

Published online by Cambridge University Press:  08 November 2000

Pere Ara ,
W. K. Nicholson  and
M. F. Yousif
Pere Ara
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, (08193) Bellaterra (Barcelona), Spain. E-mail: [email protected]
W. K. Nicholson
Affiliation:
Department of Mathematics, University of Calgary, Calgary T2N 1N4, Canada. E-mail: [email protected]
M. F. Yousif
Affiliation:
Department of Mathematics, Ohio State University, Lima, Ohio 45804, USA. E-mail: [email protected]
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Abstract

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A well known result of B. Osofsky asserts that if R is a left (or right) perfect, left and right selfinjective ring thenR is quasi-Frobenius. It was subsequently conjectured by Carl Faith that every left (or right) perfect, left selfinjective ring is quasi-Frobenius. While several authors have proved the conjecture in the affirmative under some restricted chain conditions, the conjecture remains open even if R is a semiprimary, local, left selfinjective ring withJ(R)^3=0. In this paper we construct a local ring R withJ(R)^3=0 and characterize when R is artinian or selfinjective in terms of conditions on a bilinear mapping from a D-D-bimodule toD , where D is isomorphic to R/J(R). Our work shows that finding a counterexample to the Faith conjecture depends on the existence of aD -D-bimodule over a division ring D satisfying certain topological conditions.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 42 , Issue 3 , September 2000 , pp. 391 - 404
Copyright
2000 Glasgow Mathematical Journal Trust