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Published online: 2008-06-30
On radicals and polynomial ringsSodnomkhorloo Tumurbat, Deolinda Isabel C. Mendes and Abish Mekei (1) National University of Mongolia, Ulaan Baatar, Mongolia
(2) Universidade da Beira Interior, Covilhã, Portugal
(3) National University of Mongolia, Ulaan Baatar, Mongolia
For any class ℳ of rings, it is shown that the class ℰℓ(ℳ) of all rings each non-zero homomorphic image of which contains either a non-zero left ideal in ℳ or a proper essential left ideal is a radical. Some characterizations and properties of these radicals are presented. It is also shown that, for radicals γ under certain constraints, one can obtain a strictly decreasing chain of radicals γ = γ(1) ⊃ γ(2) ⊃ ··· ⊃ γ(n) ⊃ ··· where, for each positive integer n, γ(n) is the radical consisting of all rings A such that A[ x1, … ,xn ] is in γ, thus giving a negative answer to a question posed by Gardner. Moreover, classes ℳ of rings are constructed such that there exist several such radicals γ in the interval [ ℰℓ(0),ℰℓ(ℳ) ] .
Keywords: Kurosh–Amitsur radical, essential left ideal, upper radical, polynomial rings
Tumurbat Sodnomkhorloo, Mendes Deolinda Isabel, Mekei Abish: On radicals and polynomial rings. Port. Math. 65 (2008), 261-273. doi: 10.4171/PM/1811