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Bulletin of the London Mathematical Society Advance Access originally published online on September 19, 2008
Bulletin of the London Mathematical Society 2008 40(6):917-928; doi:10.1112/blms/bdn086
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© 2008 London Mathematical Society

The Jacobson radical of rings with nilpotent homogeneous elements

Agata Smoktunowicz

Maxwell Institute of Mathematical Sciences
School of Mathematics
University of Edinburgh
Mayfield Road
Edinburgh EH9 3JZ
United Kingdom

Received 25 April 2008. Revision received 17 June 2008.

A result of Bergman says that the Jacobson radical of a graded algebra is homogeneous. It is shown that while graded Jacobson radical algebras have homogeneous elements nilpotent, it is not the case that graded algebras all of whose homogeneous elements are nilpotent are Jacobson radical. To contrast this, the following result of the author is slightly extended. Let R be a graded algebra generated in the degree one. If for every n, the n x n matrix algebra over R has all homogeneous elements nilpotent, then R is Jacobson radical.

2000 Mathematics Subject Classification 16N20, 16N40, 16W50.

This article was invited on the occasion of the author's award of a Whitehead Prize in 2006 from the London Mathematical Society. The work was supported by grant no. EPSRC EP/D071674/1.


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