In this note, the structure of
the semiprime (p,q) radicals is investigated. Let p(αj) and q(x) be polynomials
over the integers. An element a of an arbitrary associative ring R is called
(p,q)-regular if a ∈ p(a) ⋅ R ⋅ q(a). A ring R is (p,q)-regular if every element of R is
(p,q)-regular. It is easy to prove that (p,q)-regularity is a radical property and
also that it is a semiprime radical property (meaning that the radical of a
ring is a semiprime ideal of the ring) if and only if the constant coefficients
of p(x) and q(x) are ±1. It is shown that every (p,q)-semisimple ring is
isomorphic to a subdirect sum of rings which are either right primitive or left
primitive.
