A class of principal ideal rings arising from the converse of the Chinese remainder theorem

Abstract

LetRbe a (nonzero commutative unital) ring. IfIandJare ideals ofRsuch thatR/I⊕R/Jis a cyclicR-module, thenI+J=R. The ringsRsuch thatR/I⊕R/Jis a cyclicR-module for all distinct nonzero proper idealsIandJofRare the following three types of principal ideal rings: fields, rings isomorphic toK×Lfor the fieldsKandL, and special principal ideal rings(R,M)such thatM2=0.