Duplication of a module along an ideal
Let \(A\) be a commutative ring and let \(I\) be an ideal of \(A\). Let \(\pi : A \rightarrow A/I\) be the canonical surjection. The amalgamated duplication of \(A\) along \(I\) is the special pullback \(A \bowtie I := \pi \times_{A/I} \pi = \{ (a,a+i) \;| \;a \in A, i \in I \}\), a subring of \(A \times A\). This was introduced by D'Anna and Fontana, who studied its basic properties, and it has been applied to several topics. In this paper the authors consider some basic and homological properties of a special kind of \((A \bowtie I)\)-module called the duplication of \(M\) along \(I\), where \(M\) is an \(A\)-module. Specifically, \(M \bowtie I := \{ (m,m') \in M \times M \;| \;m-m' \in IM \}\). These results generalize some known results on amalgamated duplication of a ring along an ideal. The authors study when \(M \bowtie I\) is Noetherian, Artinian, or (Nil\(_*\))-Coherent as an \((R \bowtie I)\)-module. They also characterize when \(M \bowtie I\) is a (maximal) Cohen-Macaulay \((R \bowtie I)\)-module.
scientific article; zbMATH DE number 6850199
scientific article