On radicals and polynomial rings

Tumurbat, Sodnomkhorloo; Mendes, Deolinda Isabel C.; Mekei, Abish

doi:10.4171/PM/1811

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Portugaliae Mathematica

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Volume 65, Issue 2, 2008, pp. 261–273

DOI: 10.4171/PM/1811

Published online: 2008-06-30

On radicals and polynomial rings

Sodnomkhorloo Tumurbat^[1], Deolinda Isabel C. Mendes^[2] and Abish Mekei^[3]

(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 ℰ_ℓ(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, … ,x_n ] 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 [ ℰ_ℓ(0),ℰ_ℓ(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