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.
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.
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