Linkage of ideals in integral domains
Let \(R\) be a commutative ring with identity. Recall the following definitions: (1) Let \(I,J\) and \(A\) be ideals of \(R\). Then \(I\) and \(J\) are linked over \(A\) if \(A\subseteq I\cap J\), \(I=(A:J)\) and \(J=(A:I)\). (2) \(R\) has the linkage property if any two distinct nonzero ideals of \(R\) are linked. (3) \(R\) has the finite linkage property if any two distinct nonzero finitely generated ideals of \(R\) are linked. (4) \(R\) has the primary linkage property if any two distinct nonzero primary ideals of \(R\) are linked. The main results of this paper are: (1) A domain has the finite linkage property (resp. linkage property) if and only if it is Prüfer (resp. almost Dedekind). (2) A Noetherian domain (resp. one-dimensional locally Noetherian domain) has the primary linkage property if and only if it is Dedekind (resp. almost Dedekind). Many interesting examples are given.
scientific article; zbMATH DE number 7374844
scientific article