Precovers and Preenvelopes by Phantom and Ext-Phantom Morphisms

Abstract

A morphism f: M → N of left R-modules is called a phantom morphism if the induced morphism for every (finitely presented) right R-module A. Similarly, a morphism g: M → N of left R-modules is said to be an Ext-phantom morphism if the induced morphism for every finitely presented left R-module B. We prove that every left R-module has a phantom preenvelope if R is a right coherent ring and every left R-module has an Ext-phantom cover if R is a left coherent ring. In addition, we investigate the properties of precovers and preenvelopes by phantom and Ext-phantom morphisms under change of rings.