The upper nilradical and Jacobson radical of semigroup graded rings

• Ryszard Mazurek ^[3] ; Pace P. Nielsen ^[1] ; Michał Ziembowski ^[2]
  1. [1] Brigham Young University

     Brigham Young University

     Estados Unidos

     
  2. [2] Warsaw University of Technology

     Warsaw University of Technology

     Warszawa, Polonia

     
  3. [3] Bialystok University of Technology

     Bialystok University of Technology

     Białystok, Polonia

     

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• Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 219, Nº 4 ((April 2015) ), 2015, págs. 1082-1094
• Idioma: inglés
• DOI: 10.1016/j.jpaa.2014.05.036
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• Resumen

  • Given a semigroup S , we prove that if the upper nilradical Nil⁎(R)Nil⁎(R) is homogeneous whenever R is an S-graded ring, then the semigroup S must be cancelative and torsion-free. In case S is commutative the converse is true. Analogs of these results are established for other radicals and ideals. We also describe a large class of semigroups S with the property that whenever R is a Jacobson radical ring graded by S, then every homogeneous subring of R is also a Jacobson radical ring. These results partially answer two questions of Smoktunowicz. Examples are given delimiting the proof techniques.