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

Abstract

Let R be a (nonzero commutative unital) ring. If I and J are ideals of R such that R/I⊕R/J is a cyclic R-module, then I+J=R. The rings R such that R/I⊕R/J is a cyclic R-module for all distinct nonzero proper ideals I and J of R are the following three types of principal ideal rings: fields, rings isomorphic to K×L for the fields K and L, and special principal ideal rings (R,M) such that M2=0.