Further results on finitely generated projective modules
Let \(p(R)\) be the category of finitely generated projective \(R\)-modules. The author obtains the following results: (1) For any exchange ring \(R\) and any two-sided ideal \(I\) of \(R\), the canonical ring homomorphism from \(R\) to \(R/I\) induces an epimorphic group homomorphism from \(K_0(R)\) to \(K_0(R/I)\), the kernel of which is \(K=\{[P]-[Q]:P,Q\) in \(p(R)\) and \(P=PI\), \(Q=QI\}\), and \(K\) is the subgroup of \(K_0(R)\) generated by the subset \(\{[eR]\) in \(K_0(R):ee=e\) in \(I\}\) in \(K_0(R)\); (2) For any exchange ring \(R\) with primitive factor rings Artinian, there is an order preserving group isomorphism from \((K_0(R),[R])\) to \((G,u)\), where \(G\) is the Abelian group generated by the cone consisting of all the rational functions over the set of all primitive ideals of \(R\).
categories of finitely generated projective modules
scientific article; zbMATH DE number 1885822