How To Make Ext Vanish

. We describe a general construction of a module A from a given module B such that Ext (B, A) = 0 and we apply it to answer several questions on splitters, cotorsion theories, and saturated rings. 1. Introduction The main theorem, Theorem 2, gives a method for constructing from any module B, a related module A such that Ext (B, A) = 0. The fact that the module A is the union of a chain A#: ## such that for all #, A#+1/A # is isomorphic to B allows one to control the properties of A. We exploit this in order to settle some implicit and explicit questions about almost-free splitters (Theorem 7), almost cotorsion groups (Corollary 5), cotorsion theories (Theorem 10) and saturated rings (Theorem 13). We also give a su#cent condition for flat covers to exist (Corollary 11). The inspiration for Theorem 2 was the specific construction used by Gobel and Shelah in 8 to solve a long-standing problem about splitters, that is abelian groups A such that Ext (A, A) = 0. In section 3 we s. . .