Semi-local localizations of rings and subdirect decompositions of modules
The main purpose of the paper is the development of a non-commutative version of the classic Jonsson-Reid theorem which asserts that a torsionfree abelian group of finite rank is quasi-isomorphic to a finite direct sum of strongly indecomposable torsionfree abelian groups. The key concept that replaces the notion of quasi-isomorphism is the following: A pair of monomorphisms \(f: A\to A'\) and g: A'\(\to A\) in an additive category \({\mathcal M}\) is called a margimorphism pair in \({\mathcal M}\), and A and A' are called margimorphic, if fg and gf are regular maps, that is, nonzero divisors in \({\mathcal M}\). A direct sum \(B\oplus B'\) which is margimorphic to A is called a marginal decomposition of A. For an object A of the module category \({\mathcal M}\) whose endomorphism ring \(End_{{\mathcal M}}(A)\) has a classical right quotient ring \(Q_ A\), the finitely generated projective \(Q_ A\)-modules are described in terms of marginal decompositions of \(A^ n\). A major portion of the article is devoted to the development of the category theoretical machinery concerning margimorphisms, and many results are stated for additive categories.
finitely generated projective \(Q_ A\)-modules
scientific article; zbMATH DE number 4012827
scientific article