On the Dimension of Modules and Algebras (III): Global Dimension

Abstract

Let Λ be a ring with unit. If A is a left Λ-module, the dimension of A (notation: 1.dimΛ A) is defined to be the least integer n for which there exists an exact sequence 0 → Xn → … → X0 → A → 0 where the left Λ-modules X 0, …, Xn are projective. If no such sequence exists for any n, then 1. dimA A = ∞. The left global dimension of Λ is 1. gl. dim Λ = sup 1. dimA A where A ranges over all left Λ-modules, The condition 1. dimA A < n is equivalent with (A, C) = 0 for all left Λ-modules C. The condition 1.gl. dim Λ < n is equivalent with = 0. Similar definitions and theorems hold for right Λ-modules.