\(K\)-theoretically simple von Neumann regular rings
The authors investigate the differences between simplicity of a von Neumann regular ring and simplicity of its ordered Grothendieck group \(K_0\). After giving preliminaries in \S 1, they derive properties of pseudo-rank functions on ideals in a regular ring. In \S 3 they prove that if \(R\) is a stably finite, \(K_0\)-simple, non-Artinian regular ring and if the center of \(R\) is uncountable, then \(R\) is a simple unit-regular ring. They also prove that if \(R\) is a stably finite regular ring without nonzero Artinian homomorphic images then \(R\) is simple if and only if \(R\) satisfies approximate weak \(N^*\)-comparability. The authors show that given a countable field \(F\) and a countable dimension group \((G,u)\) with order-unit, there exists a stably finite, \(K_0\)-simple, regular \(F\)- algebra \(R\) such that (i) \(R\) has unique minimal nonzero ideal \(J\) and the lattice of ideals of \(R/J\) is isomorphic to the lattice of ideals of \(G\), (ii) The center of \(R\) is \(F\) and (iii) \(R\) has a unique rank function. Finally in \S 4, they prove that if \(R\) is a simple regular ring, then \(R\) satisfies weak comparability if and only if the category \(FP(R)\) of all finitely generated projective \(R\)-modules is strictly unperforated.
finitely generated projective \(R\)-modules
scientific article; zbMATH DE number 783882