On polynomial and multiplicative radicals

Abstract

We show that polynomial and multiplicative
radicals in [1] are special cases of radicals defined by means of elements. We
scrutinize the way of defining a radical γG
by a subset G of polynomials in
noncommuting indeterminates. Defining polynomial radicals, Drazin and Roberts
[1] required that the set G be closed
under composition of polynomials, and for multiplicative radicals two further
conditions were demanded. We impose milder (in fact, necessary and sufficient)
conditions, and call the so obtained radicals as weak polynomial and weak
multiplicative radicals. The Baer (prime) radical is an example for a weak
multiplicative (and so weak polynomial) radical which is not a polynomial (and
so not a multiplicative) radical. The radical class of all rings A for which the polynomial ring A[x]
is a Brown-McCoy radical ring (see [3]) is characterized in terms of
commutators, and is shown to be a multiplicative radical.

Mathematics Subject
Classification (2000): 16N80.
Key words: Hoehnke, Kurosh-Amitsur, polynomial,
multiplicative, special, Baer, nil and Brown-McCoy radical.

Quaestiones Mathematicae 26(2003), 453–469