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.