The Ohm type properties for finitely generated multiplication ideals
Let R be a commutative ring with identity. An ideal I is called a multiplication ideal if for each ideal \(J\subset I\), \(J=KI\) for some ideal K. It is known that a finitely generated ideal is a multiplication ideal if and only if it is locally principal and that a multiplication ideal is finitely generated if and only if it satisfies the cancellation condition AI\(\subseteq BI\) implies \(A\subseteq B+Ann(I).\) The purpose of this paper is to investigate the two ``Ohm type'' properties: \((1)\quad A^ n=B^ n\) for some n implies \(A=B\) (A and B ideals) and \((2)\quad (A:B)^ n=(A^ n:B^ n).\) For example, it is shown that if A and B are finitely generated multiplication ideals with \(A+B\) also a multiplication ideal, then \((1)\quad A^ n=B^ n\) for some n implies that \(A+Ann(A+B)=B+Ann(A+B)\) and \((2)\quad (A:B)^ n+Ann(B^ n)=(A^ n:B^ n)+Ann(B^ n)\) for all n.
scientific article; zbMATH DE number 4021290