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.
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机译：Bergman的结果说，渐变代数的Jacobson根是齐次的。结果表明，虽然渐变的Jacobson根代数具有齐次幂的幂元素，但并非所有均质元素都是零次幂的渐变代数都是Jacobson根。与此相反，作者的以下结果略有扩展。令R为一阶生成的渐变代数。如果对于每个n，R上的n x n矩阵代数都具有所有齐次幂的齐次元素，则R是Jacobson根。
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