THE IDEALS OF AN IDEAL EXTENSION

Given two unital associative rings R ⊆ S, the ring S is said to be an ideal (or Dorroh) extension of R if S = R ⊕ I, for some ideal I ⊆ S. In this note, we investigate the ideal structure of an arbitrary ideal extension of an arbitrary ring R. In particular, we describe the Jacobson and upper nil radicals of such a ring, in terms of the Jacobson and upper nil radicals of R, and we determine when such a ring is prime and when it is semiprime. We also classify all the prime and maximal ideals of an ideal extension S of R, under certain assumptions on the ideal I. These are generalizations of earlier results in the literature.