International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 19607, 5 pages
doi:10.1155/IJMMS/2006/19607
A class of principal ideal rings arising from the converse of the
Chinese remainder theorem
Department of Mathematics, University of Tennessee, Knoxville 37996-1300, TN, USA
Received 13 February 2006; Revised 4 May 2006; Accepted 9 May 2006
Copyright © 2006 David E. Dobbs. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.