Structure of semiprime (p, q) radicals

Pseudoregular radical classes

“…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]).…”

“…In general we have the following, which may be well-known and in a more general form, but we include its proof anyway. PROOF: NOW we have that p = dp 1 Thus an (x | g)-pseudoregular class (which is always radical, being a (q; l)-regular radical class) is semiprime if and only if |g(0)| = 1. "On the other hand, there are pseudoregular classes TZ Ptq with |g(0)| -1 which are not even radical classes.…”

A class of regularities for rings

“…and q are arbitrary, but fixed integral polynomials. This regularity is treated by Goulding and Ortiz (1971), Musser (1971), Ortiz (1971) and McKnight and Musser (1972).…”

“…All regularities introduced up to then were regularities in the sense of it. However, in 1971 a wide 437 class of regularities was introduced by McKnight and others, called (/», ^-regularities, and it was noted by Goulding and Ortiz (1971) that some of these (p,q)regularities fail to satisfy the set of axioms of the theory of Brown and McCoy. After these introductory historical remarks we now come to a brief description of the substance of this paper. In Chapter 2 we introduce a general concept of regularities for rings.…”

Radical Classes Closed Under Products

“…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…”