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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2020; 60(4): 711-721

Published online December 31, 2020

### On n-Amitsur Rings

Baatar Ochirbat, Deolinda I. C. Mendes* ,Sodnomkhorloo Tumurbat

Department of Mathematics, Mongolian University of Science and Technology, Ulaanbaatar, Mongolia
e-mail : ochirbat@must.edu.mn
Department of Mathematics, University of Beira Interior, Portugal
e-mail : imendes@ubi.pt
Department of Mathematics, National University of Mongoliaand School of Applied Science and Technology, Ulaanbaatar, Mongolia
e-mail : stumurbat@hotmail.com

Received: August 25, 2019; Revised: July 18, 2020; Accepted: July 21, 2020

### Abstract

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