Special (p;q) Radicals

Abstract

In [3], the study of (p;q) radicals was initiated. In this paper, the integral polynomials p(x) and q(x) which determine the Jacobson radical are characterized and the Jacobson radical is shown to be the only semiprime (p;q) radical for which all fields are semisimple. Also, it is observed that the prime, nil, and Brown-McCoy radicals are not (p;q) radicals. To show that the semiprime (p;q) radicals are special and that they can be determined by subclasses of the class of primitive rings, a classification theorem for (p;q)-regular primitive rings is given. Finally, it is shown that the collection of semiprime (p;q) radicals and the collection of semiprime (p;1) radicals coincide.

Pseudoregular radical classes

“…A great many radical T2. Pi9 turn out to be J as is evident from results in [4]: if |<? (0)| = 1 then this is the case exactly when q(x) has a factor of the form (ax + 1), where a ^ 0 and for each prime divisor m of a, there is an integer n such that m divides p(n).…”

“…Here is an example to show that in general a semiprime pseudoregular radical class need not. Let f{x,y) = x 2 + (x 4 -x 2 )y, so thatp(a:) = x 1 and q(x) = x 4 -i 2 ; the conditions of Theorem 1.1 are satisfied as is easily checked, so IZf is a radical class. It is easy to see that all zerorings are in 11 f, which is therefore semiprime, so if it were (p; g)-regular for some p, q, then it would contain J (by [4,Theorem 3]).…”

“…Let f{x,y) = x 2 + (x 4 -x 2 )y, so thatp(a:) = x 1 and q(x) = x 4 -i 2 ; the conditions of Theorem 1.1 are satisfied as is easily checked, so IZf is a radical class. It is easy to see that all zerorings are in 11 f, which is therefore semiprime, so if it were (p; g)-regular for some p, q, then it would contain J (by [4,Theorem 3]). However, R -Z 0 \X]/(X 3 ) is nil and not in 1Z f , since (writing li for the image of X in R), there would need to be a P(X) € R such that X -X P(X) = 0, which there is not.…”

Radical Classes Closed Under Products

“…The classes of (ρ; σ)-regular rings introduced by McKnight and Musser [58] are product closed radical classes.…”

“…For every ρ, σ, the class R ρσ of (ρ; σ)-regular rings is a radical class. The idea seems to be due to McKnight and is studied in [42], [58], [59], [60]. In these papers the polynomial functions…”