# On radicals and polynomial rings

### Sodnomkhorloo Tumurbat

National University of Mongolia, Ulaan Baatar, Mongolia### Deolinda Isabel C. Mendes

Universidade da Beira Interior, Covilhã, Portugal### Abish Mekei

National University of Mongolia, Ulaan Baatar, Mongolia

## Abstract

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 $A[ x_{1}, … ,x_{n} ]$ 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 $[ E_{ℓ}(0),E_{ℓ}(M) ]$ .

## Cite this article

Sodnomkhorloo Tumurbat, Deolinda Isabel C. Mendes, Abish Mekei, On radicals and polynomial rings. Port. Math. 65 (2008), no. 2, pp. 261–273

DOI 10.4171/PM/1811