The Jacobson radical of rings with nilpotent homogeneous elements

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.