The concepts of an Amitsur ring and a hereditary Amitsur ring, which were introduced and studied by S. Tumurbat in a recent paper, are generalized. For a positive integer n, a ring A is said to be an n-Amitsur ring if γ ( A [ X n ] ) = ( γ ( A [ X n ] ) ∩ A ) [ X n ] for all radicals γ, where A [ X n ] is the polynomial ring over A in n A satisfies the above equation for all hereditary radicals γ, then A is said to be a hereditary n-Amitsur ring. Characterizations and examples of these rings are provided. Moreover, new radicals associated with n-Amitsur rings are introduced and studied. One of these is a special radical and its semisimple class is polynomially extensible.

Keywords: Amitsur rings, hereditary Amitsur rings, radicals, radicals with the Amitsur property