Vol. 37, No. 1, 1971

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Structure of semiprime (p, q) radicals

Thomas L. Goulding and Augusto H. Ortiz

Vol. 37 (1971), No. 1, 97–99

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.

Mathematical Subject Classification
Primary: 16A21
Received: 2 April 1970
Published: 1 April 1971
Thomas L. Goulding
Augusto H. Ortiz