Linear semiprime (p;q) radicals

Abstract

This paper introduces McKnight's (p; gO-regularity and (p; q) radicals, a collection of radicals which contains the Jacobson radical and the radicals of regularity and strong regularity among its members. The linear semiprime (p; q) radicals are classified canonically and, as a result of this classification, these radicals can be distinguished by the fields GF(p) and are shown to form a lattice. The semiprime (p; q) radicals are found to be hereditary and the linear semiprime (p; q) radical of a complete matrix ring of a ring R is determined to be the complete matrix ring over the (p; q) radical of R. More generally, the (p; q) radical of a complete matrix ring over R is contained in the matrix ring over the (p; q) radical of R for all (p; q) radicals.

Special (p;q) Radicals

“…An element r Ç R is (p;q)-regular if r G p(r)Rq(r), and R is a (p;q)-regular ring if every element of R is (p;q)-regular. In [3], it is shown that every associative ring R contains a largest (p\q)-regular ideal (p(x)Rq(x)) and that the function which assigns the ideal (p(x)Rq(x)) to the ring R is a radical function in the sense of Amitsur and Kurosh. Moreover, A. H. Ortiz has shown (see [3]) that a (p;q) radical is semiprime (contains the prime radical) if and only if the constant terms of p(x) and q(x) are ±1; the Jacobson radical J(R) is the semiprime (p',q) radical given by ((x + 1)R)-It is easy to see that all semiprime (p',q) radicals are hereditary [3] and, hence, supernilpotent.…”

“…Introduction. 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.…”

Pseudoregular radical classes

“…It is easy to give sufficient conditions for hereditariness of 11 f. These are the relevant version of conditions given in [5], which they generalise in the (q; l)-regular case. It is not known if a converse for this result can be obtained.…”

“…The classes of quasiregular and von Neumann regular rings have the form TZf for some / : in the former case, we may let / = x + y + xy and g -y + z + yz; in the latter, f -x -xyx and g = y + z -2xyz + xyzyx. Indeed these examples may be generalised as follows: if p, q € Z [x], then the element x -p(x)yq(x) of Z 0 [x,2/] is associating, as is implicitly shown in [5], and the resulting radical class is the class of (p;q)-regular rings. This family also includes Divinsky's D-regular radical class.…”

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…”