A NOTE ON NIL AND JACOBSON RADICALS IN GRADED RINGS

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Abstract

It was shown by Bergman that the Jacobson radical of a Z-graded ring is homogeneous. This paper shows that the analogous result holds for nil radicals, namely, that the nil radical of a Z-graded ring is homogeneous. It is obvious that a subring of a nil ring is nil, but generally a subring of a Jacobson radical ring need not be a Jacobson radical ring. In this paper, it is shown that every subring which is generated by homogeneous elements in a graded Jacobson radical ring is always a Jacobson radical ring. It is also observed that a ring whose all subrings are Jacobson radical rings is nil. Some new results on graded-nil rings are also obtained.

Original languageEnglish
Article number1350121
Number of pages8
JournalJournal of algebra and its applications
Volume13
Issue number4
DOIs
Publication statusPublished - Jun 2014

Keywords

  • Graded rings
  • nil rings
  • nil radical
  • Jacobson radical
  • SEMIGROUP RINGS

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