The ideals of an ideal extension

TL;DR

This paper explores the structure of ideal extensions of rings, analyzing radicals, prime, and maximal ideals, and generalizing previous results in ring theory.

Contribution

It provides a comprehensive description of radicals and ideal classifications in ideal extensions of rings, extending earlier findings.

Findings

Describes Jacobson and upper nil radicals in ideal extensions.

Classifies prime and maximal ideals under certain conditions.

Generalizes previous results on ideal structures in ring extensions.

Abstract

Given a unital associative ring S and a subring R, we say that S is an ideal (or Dorroh) extension of R if for some ideal I of S, S = R + I, where the sum is direct. 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.