# The ideals of an ideal extension - Mathematics > Rings and Algebras

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Abstract: Given a unital associative ring S and a subring R, we say that S is an idealor Dorroh extension of R if for some ideal I of S, S = R + I, where the sumis direct. In this note we investigate the ideal structure of an arbitraryideal extension of an arbitrary ring R. In particular, we describe the Jacobsonand upper nil radicals of such a ring, in terms of the Jacobson and upper nilradicals of R, and we determine when such a ring is prime and when it issemiprime. We also classify all the prime and maximal ideals of an idealextension S of R, under certain assumptions on the ideal I. These aregeneralizations of earlier results in the literature.

Author: ** Zachary Mesyan**

Source: https://arxiv.org/