Co-stability of Radicals and Its Applications to PI-Theory
Copyright policy )Co-stability of Radicals and Its Applications to PI-Theory
We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either char F = 0 or char F > dim A, then the Jacobson radical J(A) is an H-subcomodule of A. In particular, if A is a finite-dimensional associative algebra over such a field F, graded by any group, then the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite-dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of Amitsur's conjecture for graded polynomial identities of finite-dimensional associative algebras over a field of characteristic 0, graded by any group. In addition, we provide a criterion for graded simplicity of an associative algebra in terms of graded codimensions.
- Vrije Universiteit Brussel Belgium
nilpotent radical, Hopf algebras and their applications, Graded rings and modules (associative rings and algebras), grading, Mathematics - Rings and Algebras, codimension, Jacobson radical, radicals, Hopf algebra, Jacobson radical, quasimultiplication, solvable radical, \(H\)-comodule algebra, Rings and Algebras (math.RA), FOS: Mathematics, Coalgebras and comodules; corings, polynomial identity, H-comodule algebra, H-module algebra, Structure theory for Lie algebras and superalgebras, Primary 16W50, Secondary 17B05, 16R10, 16R50, 17B70, 16T05, 16T15, Amitsur's conjecture, associative algebra, Lie Algebra
nilpotent radical, Hopf algebras and their applications, Graded rings and modules (associative rings and algebras), grading, Mathematics - Rings and Algebras, codimension, Jacobson radical, radicals, Hopf algebra, Jacobson radical, quasimultiplication, solvable radical, \(H\)-comodule algebra, Rings and Algebras (math.RA), FOS: Mathematics, Coalgebras and comodules; corings, polynomial identity, H-comodule algebra, H-module algebra, Structure theory for Lie algebras and superalgebras, Primary 16W50, Secondary 17B05, 16R10, 16R50, 17B70, 16T05, 16T15, Amitsur's conjecture, associative algebra, Lie Algebra
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