Power cancellation of modules
Let Mod-R (resp. \(Mod_ f-R)\) denote the category of unital right (resp. finitely generated) R-modules. \(X\in Mod_ f-R\) satisfies ''power cancellation'' if for all M,N\(\in Mod-R\), \(M\oplus X\cong N\oplus X\Rightarrow M^{(t)}\cong N^{(t)}\) for some \(0<t\in {\mathbb{Z}}\). Here \(M^{(t)}\) is the direct sum of t copies of M. If D is a Dedekind domain, any finitely generated module satisfies cancellation (i.e., \(t=1)\). If R is a \({\mathbb{Z}}\)-order in a separable \({\mathbb{Q}}\)-algebra, then power cancellation holds for R-lattices. It is shown in this article that for R being an integrally closed domain, \(M\oplus X\cong N\oplus X\Rightarrow M^{(t)}\cong N^{(t)}\) for some \(0<t\in {\mathbb{Z}}\) for all finitely generated S-modules M, N, X, whenever S is a module finite algebra if and only if one is in the stable range of the integral closure of R in the algebraic closure of its quotient field. In particular, this holds whenever R is a Dedekind domain with all residue fields torsion. This extends work of Goodearl, who showed this holds for module finite (and, more generally, finite rank) algebras over the integers.
scientific article; zbMATH DE number 3955063