This paper introduces
McKnight’s (p;q)-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.
