Separative cancellation for projective modules over exchange rings
An \(R\)-module \(M\) has the exchange property if for every \(R\)-module \(A\) and any decompositions \(A=M'\oplus N=\bigoplus_{i\in I}A_i\), with \(M'\cong M\), there exist submodules \(A_i'\subseteq A_i\) such that \(A=M'\oplus\left(\bigoplus_{i\in I}A_i'\right)\). If the above condition is satisfied whenever the index set \(I\) is finite, \(M\) is said to satisfy the finite exchange property. A ring \(R\) is called an exchange ring if \(R_R\) satisfies the (finite) exchange property. A ring \(R\) is called a separative ring if the following condition holds for all \(A,B\in\text{FP}(R)\): \(A\oplus A\cong A\oplus B\cong B\oplus B\Rightarrow A\cong B\), where \(\text{FP}(R)\) is the class of all finitely generated projective modules over the ring \(R\). The separativity is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. In this paper, the authors develop some basic characterizations and initial applications of separativity, study stable rank conditions on exchange rings, prove the extension theorem for separative exchange rings, and give a corresponding extension result for the smaller class of strongly separative exchange rings. The results of this paper are particularly useful for the study of ring and module theory.
scientific article; zbMATH DE number 1195644
scientific article