On radicals and polynomial rings
Sodnomkhorloo TumurbatNational University of Mongolia, Ulaan Baatar, Mongolia
Deolinda Isabel C. MendesUniversidade da Beira Interior, Covilhã, Portugal
Abish MekeiNational 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) ] .
Cite this article
Sodnomkhorloo Tumurbat, Deolinda Isabel C. Mendes, Abish Mekei, On radicals and polynomial rings. Port. Math. 65 (2008), no. 2, pp. 261–273DOI 10.4171/PM/1811