On commutativity of ideal extensions

TL;DR

This paper investigates the conditions under which ideal extensions of rings are commutative, introduces new construction methods, and classifies certain fields as quotients of such extensions.

Contribution

It provides a novel construction of noncommutative rings with specific ideal properties and classifies fields obtainable as quotients of these rings.

Findings

Constructed a noncommutative ring with a central, idempotent ideal leading to a field quotient.

Answered an open question regarding the existence of such rings.

Classified fields of characteristic 0 that can be realized as quotients of ideal extensions.

Abstract

In this paper we examine the commutativity of ideal extensions. We introduce methods of constructing such extensions, in particular we construct a noncommutative ring T which contains a central and idempotent ideal I such that T/I is a field. This answers a question from [2]. Moreover we classify fields of characteristic 0 which can be obtained as T/I for some T.